🕷️ Crawler Inspector

URL Lookup

Direct Parameter Lookup

Raw Queries and Responses

1. Shard Calculation

Query:

Response:

Calculated Shard: 2 (from laksa114)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa002.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical, ml_categories_json AS ml_categories_json, ml_types_json AS ml_types_json, ml_intent_types_json AS ml_intent_types_json, meta_language AS meta_language, attrs_author AS attrs_author, ifNull(toUnixTimestamp(attrs_publish_time), 0) AS attrs_publish_time, ifNull(toUnixTimestamp(attrs_original_publish_time), 0) AS attrs_original_publish_time, ifNull(attrs_is_republished, 0) AS attrs_is_republished, ifNull(attrs_nr_words, 0) AS attrs_nr_words, ifNull(attrs_boilerpipe_nr_words, 0) AS attrs_boilerpipe_nr_words, ifNull(body_ext_links_number, 0) AS body_ext_links_number, ifNull(body_int_links_number, 0) AS body_int_links_number, ifNull(meta_nofollow, 0) AS meta_nofollow, ifNull(meta_noarchive, 0) AS meta_noarchive, ifNull(props_was_rendered, 0) AS props_was_rendered, ifNull(src_redirect, \'\') AS src_redirect, ifNull(download_time_msec, 0) AS download_time_msec, ifNull(download_ttfb_msec, 0) AS download_ttfb_msec, ifNull(download_size, 0) AS download_size FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://www.symbolab.com/solver/partial-derivative-calculator\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.symbolab.com/solver/partial-derivative-calculator\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/www.symbolab.com\/solver\/partial-derivative-calculator","crawl_time":1775102728,"first_indexed_time":1421694210,"http_code":200,"src_unparsed":"com,symbolab!www,\/solver\/partial-derivative-calculator s443","src_root_hash":"1977249391401099202","history_drop_reason":null,"meta_title":"Partial Derivative Calculator","meta_descriptions":["Free partial derivative calculator - partial differentiation solver step-by-step"],"attrs_boilerpipe_text":"A Step-by-Step Guide to Partial Derivatives Calculator\nMany moments in daily life hinge on more than one changing factor. The taste of homemade soup, for example, is not just about the mix of spices but also the time the pot spends simmering. A perfect photograph is shaped by both the\nangle\nof the light and the split-second timing of the shutter. There’s rarely only one ingredient at play. When several\nvariables\ncome together to shape an outcome, partial derivatives offer a quiet way to notice: what happens if only one piece of the puzzle moves, while everything else stands still? In this article, partial derivatives will be explored one careful step at a time—what they are, why they matter, how they show up in daily life, and how to work with them using Symbolab’s Partial Derivative Calculator.\nWhat are Partial Derivatives\nThink of any outcome that depends on a handful of choices, not just one. In math, these relationships are described by\nfunctions\nlike $f(x, y)$, where both $x$ and $y$ are inputs—sometimes these are hours studied and hours slept, sometimes they’re temperature and humidity, sometimes they’re the amount of flour and the oven’s heat. Each factor quietly pushes and pulls the result in its own way.\nA partial derivative asks: if one variable changes—say, $x$—while $y$ stays exactly the same, how does the outcome respond? It’s like gently turning one dial on a stereo and listening for just that note in the music. Mathematically, the partial derivative with respect to $x$ is written as:\n$\\frac{\\partial f}{\\partial x}$\nand the same question about $y$ gives:\n$\\frac{\\partial f}{\\partial y}$\nNo need to let the symbols overwhelm; the heart of it is this: only one input shifts, the rest are frozen in place.\nFor something even more down-to-earth, think about cooking rice. The final texture, $T$, depends on both water $w$ and heat $h$—so $T(w, h)$. Suppose you want to know how the rice’s texture changes when you increase the heat while keeping the water level constant. That rate of change is captured by the partial derivative with respect to heat, written as:\n$\\frac{\\partial T}{\\partial h}$\nmeasures just that. How does the rice change, with water unchanged, as the flame turns up?\nThis idea isn’t only for kitchens or science labs; it appears wherever complex outcomes depend on several choices at once. Partial derivatives let us slow down and focus, revealing how small, single changes can shape a bigger story. They’re a first, essential step toward making sense of the whole beautiful tangle of variables that fill real life.\nUnderstanding Multivariable Functions\nIt’s rare for one single variable to hold all the power. Most real-world outcomes—the brightness of a room, the total at a checkout counter, the time it takes to bike across town—depend on two or more factors working together. In math, multivariable functions capture these relationships.\nA multivariable function might look like $f(x, y)$, where both $x$ and $y$ influence the result. Think about a weather app that predicts temperature based on time of day ($t$) and cloud cover ($c$), or a bakery’s profit, $P(c, p)$, that depends on the number of cakes ($c$) and pies ($p$) sold. Each variable tugs on the outcome in its own way.\nFor example:\nIn a classroom, a student’s test score might depend on hours spent studying ($s$) and hours slept the night before ($h$): $S(s, h)$.\nA family’s grocery bill could be modeled as $B(f, v)$, where $f$ is pounds of fruit and $v$ is pounds of vegetables in the cart.\nSometimes, these functions are simple sums:\n$f(x,y)=2x+3y$\nOther times, they weave together in more interesting patterns, like\n$f(x, y) = x^2 + xy + y^2$\nor\n$f(x, y, z) = xyz$\nMultivariable functions create a kind of landscape—each combination of inputs gives a new point in that landscape. Understanding how each variable shapes the outcome, one at a time, is where partial derivatives enter the scene. But before calculating changes, it helps to pause and notice: in math as in life, the most interesting results often depend on more than one ingredient.\nThe Concept of Partial Derivatives\nPicture a hill stretching out in every direction. The height at any point depends on how far north or east you walk. This is a classic multivariable function—height as a function of $x$ (east-west) and $y$ (north-south): $h(x, y)$. Now, imagine standing still in the north-south direction, and taking a small step east. The question becomes: how does the height change as you move only east, with your north-south position unchanged?\nThat is the spirit of a partial derivative. It measures how a function’s output shifts as you change one input ever so slightly, while all other variables are frozen. In mathematical language, for a function $f(x, y)$, the partial derivative with respect to $x$ tells the rate at which $f$ increases or decreases if only $x$ is allowed to change:\n$\\frac{\\partial f}{\\partial x}$\nIn the world beyond math, this might look like asking: How does a plant’s growth respond to extra sunlight if water stays constant? Or, how does the comfort in a room shift if only the temperature rises, while humidity does not budge?\nThe core idea is focus—tuning out every other influence for a moment and seeing the effect of just one change. It’s a small, controlled experiment, repeated for each variable in turn. Partial derivatives offer a way to trace the shape of a complex system, one direction at a time, revealing how each thread is woven into the whole.\nNotation and Mathematical Definition of Partial Derivatives\nAt first, the notation for partial derivatives might look unfamiliar, but each symbol carries a clear meaning. The most common is the “curly d,” written as $\\partial$, which signals that only one variable is changing at a time.\nFor a function $f(x, y)$, the partial derivative with respect to $x$ is written as:\n$\\frac{\\partial f}{\\partial x}$\nThis is read as “the partial derivative of $f$ with respect to $x$.” Here, $y$ is held constant—almost as if $y$ has been pressed on pause.\nThe process works the same for $y$:\n$\\frac{\\partial f}{\\partial y}$\nAnd if there are more variables, the pattern continues. For $f(x, y, z)$, you might see:\n$\\frac{\\partial f}{\\partial z}$\nThe Formal Definition\nMathematically, the partial derivative is defined as a limit. For $f(x, y)$, the partial derivative with respect to $x$ at a specific point $(a, b)$ is:\n$\\displaystyle \\frac{\\partial f}{\\partial x}(a,b) = \\lim_{h \\to 0} \\frac{f(a+h,,b) - f(a,,b)}{h}$\nThis formula captures what happens to $f$ as $x$ increases by a tiny amount $h$, while $y$ remains fixed at $b$.\nNo matter how many variables there are, the logic is always the same: only one is allowed to move, the rest are temporarily held steady. With practice, these symbols become a way to trace the changing shape of any landscape, mathematical or otherwise.\nHigher-Order and Mixed Partial Derivatives\nHigher-Order Partial Derivatives\nA higher-order partial derivative is found by taking the partial derivative of a partial derivative. For example, take a function $f(x, y)$. The first partial derivative with respect to $x$ is $\\frac{\\partial f}{\\partial x}$. If this is differentiated again with respect to $x$, the result is the second-order partial derivative:\n$\\frac{\\partial^2 f}{\\partial x^2}$\nThis measures how the rate of change with respect to $x$ changes as $x$ continues to change. Imagine climbing a hill: the second-order derivative tells if the slope is getting steeper or more gentle as you move east.\nMixed Partial Derivatives\nMixed partial derivatives measure how the rate of change with respect to one variable shifts as another variable changes. For $f(x, y)$, the mixed partial derivative with respect to $x$ and then $y$ is:\n$\\frac{\\partial^2 f}{\\partial y \\partial x}$\nThis means: first, find the partial derivative with respect to $x$, then take the partial derivative of that result with respect to $y$. In real life, this might answer questions like, “How does the effect of more study hours on a test score change as sleep hours increase?”\nIf the function is well-behaved (meaning it is “nice enough,” with no sharp corners or breaks), the mixed partial derivatives can be taken in any order and still get the same answer:\n$\\frac{\\partial^2 f}{\\partial y \\partial x} = \\frac{\\partial^2 f}{\\partial x \\partial y}$\nInterpreting Partial Derivatives\nInterpreting a partial derivative means understanding what the value represents in a specific context. For a function $f(x, y)$, the partial derivative $\\frac{\\partial f}{\\partial x}$ describes how much $f$ will increase or decrease if $x$ changes by a tiny amount, while $y$ is held constant.\nFor example, if $f(x, y)$ gives the amount of money earned at a lemonade stand, where $x$ is the number of lemons used and $y$ is the cups of sugar, then $\\frac{\\partial f}{\\partial x}$ tells how much extra money is earned for each extra lemon added—assuming the sugar stays the same. This makes it possible to see, one step at a time, how a single choice affects the result.\nReal-World Applications of Partial Derivatives\nPartial derivatives might seem abstract at first glance, but their fingerprints are everywhere in daily life and across nearly every field that measures, predicts, or designs.\nIn economics, $\\frac{\\partial P}{\\partial n}$ measures how profit changes with each additional product sold, keeping advertising constant.\nIn medicine, $\\frac{\\partial R}{\\partial d}$ shows how a patient’s response varies with drug dosage, while sleep remains fixed.\nIn engineering, $\\frac{\\partial S}{\\partial T}$ reveals how material stress shifts with temperature, force held steady.\nIn weather forecasting, $\\frac{\\partial T}{\\partial h}$ tells how temperature changes with altitude at a given location.\nIn environmental science, $\\frac{\\partial C}{\\partial t}$ describes how pollution concentration changes over time at one spot.\nIn machine learning, $\\frac{\\partial L}{\\partial w}$ helps algorithms adjust model weights to improve prediction accuracy.\nHow to Calculate Partial Derivatives Manually\nSolving partial derivatives by hand is often about careful focus—paying attention to one variable, while treating all others as if they’re constants. Here’s how to approach it, step by step:\n1. Pick the Variable\nDecide which variable you’re differentiating with respect to—call it $x$. All other variables (like $y$, $z$, etc.) are held constant during this step.\n2. Differentiate as Usual\nApply the rules of differentiation to $x$, just as you would for a single-variable function. If you see a $y$ or $z$, treat it as if it were just a number.\n3. Leave Other Variables Alone\nVariables that aren’t the one you’re differentiating by don’t change. If the term doesn’t contain $x$, its\nderivative\nwith respect to $x$ is $0$.\nExample\nSuppose $f(x, y) = 3x^2y + 2y^2 + 7x$.\nTo find the partial derivative with respect to $x$:\n$3x^2y$ becomes $6xy$ (since $y$ is treated as constant)\n$2y^2$ becomes $0$ (no $x$ in this term)\n$7x$ becomes $7$\nSo,\n$\\frac{\\partial f}{\\partial x} = 6xy + 7$\nTo find the partial derivative with respect to $y$:\n$3x^2y$ becomes $3x^2$ (since $x$ is treated as constant)\n$2y^2$ becomes $4y$\n$7x$ becomes $0$\nSo,\n$\\frac{\\partial f}{\\partial y} = 3x^2 + 4y$\nThis approach works for any function of two or more variables. Keep your focus on just one variable at a time, and the process becomes much clearer. Calculating partial derivatives manually not only builds understanding, but also strengthens confidence for more complex problems.\nCommon Mistakes and How to Avoid Them\nPartial derivatives are powerful, but a few classic missteps can trip up even careful students. Here are some common mistakes—plus gentle reminders to help keep your work on track:\nForgetting to Hold Other Variables Constant:\nWhen differentiating with respect to $x$, every other variable (like $y$ or $z$) should be treated as if it is a fixed number. Only $x$ is “alive” in that moment.\nDifferentiating Terms Without the Chosen Variable:\nAny term that does not contain the variable you are differentiating with respect to has a derivative of $0$ for that step. For example, in $\\frac{\\partial}{\\partial x}(5y^2)$, the answer is $0$.\nMixing Up Notation:\nUsing $\\frac{d}{dx}$ instead of $\\frac{\\partial}{\\partial x}$, or forgetting the curly $\\partial$, is a small but common slip. Remember, partial derivatives always use the $\\partial$ symbol.\nDropping Variables Too Early:\nDon’t remove variables from your answer unless the differentiation process sets them to $0$. For example, $3x^2y$ differentiated with respect to $x$ becomes $6xy$, not just $6x$.\nForgetting Which Variable is “In Play”:\nAlways double-check which variable you’re differentiating by, especially in functions with several variables.\nHow to Use the Partial Derivative Calculator on Symbolab\nWhen calculations get lengthy or you want extra reassurance, Symbolab’s Partial Derivative Calculator is a supportive tool for every learner. Here’s how to use it, step by step:\nStep 1: Enter the Expression\nType directly using your keyboard: Enter your function, such as $f(x, y) = 3x^2y + 2y^2 + 7x$.\nUse the math keyboard: Handy for entering square roots, fractions,\nexponents\n, and other special math symbols.\nUpload a photo: Snap a picture of a handwritten problem or textbook page.\nUse the Chrome extension: Capture math directly from a webpage by highlighting and screenshotting the expression.\nPress the “Go” button.\nStep 2: View Step-by-Step Breakdown\nWatch as each part of the solution is explained clearly, with the option to go through one step at a time.\nIf a step doesn’t make sense, or if you’re curious about a detail, use the ‘Chat with Symbo’ to ask questions or get clarifications tailored to your needs.\nConclusion\nPartial derivatives are more than a set of rules—they are a way of seeing how each piece of a bigger puzzle contributes to change. Whether solved by hand or explored with Symbolab’s calculator, each calculation reveals a little more about how the world’s variables interact. With patience and practice, these tools help make complex situations understandable, guiding thoughtful decisions in science, business, engineering, or everyday life.\nFrequently Asked Questions (FAQ)\nHow do you find the partial derivative?\nTo calculate the partial derivative of a function choose the variable with respect to which you want to take the partial derivative, and treat all the other variables as constant. Differentiate the function with respect to the chosen variable, using the rules of differentiation.\nWhat are the rules of partial derivatives?\nPartial derivatives follow the sane rules as derivatives: the sum rule, the difference rule, the product rule, the quotient rule, and the chain rule.\nWhat is the sum rule of partial derivatives?\nThe sum rule of partial derivatives is a technique for calculating the partial derivative of the sum of two functions. It states that if f(x,y) and g(x,y) are both differentiable functions, then:\n∂(f+g)\/∂x = ∂f\/∂x + ∂g\/∂x\n∂(f+g)\/∂y = ∂f\/∂y + ∂g\/∂y\nWhat is the difference rule of partial derivatives?\nThe difference rule of partial derivatives is a technique for calculating the partial derivative of the difference of two functions. It states that if f(x,y) and g(x,y) are both differentiable functions, then:\n∂(f-g)\/∂x = ∂f\/∂x - ∂g\/∂x\n∂(f-g)\/∂y = ∂f\/∂y - ∂g\/∂y\nWhat is the product rule of partial derivatives?\nThe product rule of partial derivatives is a technique for calculating the partial derivative of the product of two functions. It states that if f(x,y) and g(x,y) are both differentiable functions, then:\n∂(fg)\/∂x = f∂g\/∂x + g∂f\/∂x\n∂(fg)\/∂y = f∂g\/∂y + g∂f\/∂y\nWhat is the quotient rule of partial derivatives?\nThe quotient rule of partial derivatives is a technique for calculating the partial derivative of the quotient of two functions. It states that if f(x,y) and g(x,y) are both differentiable functions and g(x,y) is not equal to 0, then:\n∂(f\/g)\/∂x = (∂f\/∂xg - f∂g\/∂x)\/g^2\n∂(f\/g)\/∂y = (∂f\/∂yg - f∂g\/∂y)\/g^2\nWhat is the chain rule of partial derivatives?\nThe chain rule of partial derivatives is a technique for calculating the partial derivative of a composite function. It states that if f(x,y) and g(x,y) are both differentiable functions, and y is a function of x (i.e. y = h(x)), then: ∂f\/∂x = ∂f\/∂y * ∂y\/∂x\nWhat is the partial derivative of a function?\nThe partial derivative of a function is a way of measuring how much the function changes when you change one of its variables, while holding the other variables constant.","attrs_markdown":"You can see your coupon in the [user page](https:\/\/www.symbolab.com\/user#subscription)\n\n[Go To QuillBot Upgrade to Pro Continue to site]()\n\n[We've updated our]() [Privacy Policy](https:\/\/www.symbolab.com\/privacy-policy) effective December 15. Please read our updated Privacy Policy and tap\n\nContinue\n\n- [Solutions](https:\/\/www.symbolab.com\/solver)\n  [Integral Calculator](https:\/\/www.symbolab.com\/solver\/integral-calculator) [Derivative Calculator](https:\/\/www.symbolab.com\/solver\/derivative-calculator) [Algebra Calculator](https:\/\/www.symbolab.com\/solver\/algebra-calculator) [Matrix Calculator](https:\/\/www.symbolab.com\/solver\/matrix-calculator) [More...](https:\/\/www.symbolab.com\/solver)\n- [Graphing](https:\/\/www.symbolab.com\/graphing-calculator)\n  [Line Graph Calculator](https:\/\/www.symbolab.com\/graphing-calculator\/line) [Exponential Graph Calculator](https:\/\/www.symbolab.com\/graphing-calculator\/exponential-graph) [Quadratic Graph Calculator](https:\/\/www.symbolab.com\/graphing-calculator\/quadratic-graph) [Sine Graph Calculator](https:\/\/www.symbolab.com\/graphing-calculator\/sine-graph) [More...](https:\/\/www.symbolab.com\/graphing-calculator#examples=true)\n- [Calculators](https:\/\/www.symbolab.com\/calculators)\n  [BMI Calculator](https:\/\/www.symbolab.com\/calculator\/fitness\/bmi) [Compound Interest Calculator](https:\/\/www.symbolab.com\/calculator\/finance\/compound_interest_rate) [Percentage Calculator](https:\/\/www.symbolab.com\/calculator\/math\/percentage) [Acceleration Calculator](https:\/\/www.symbolab.com\/calculator\/physics\/acceleration) [More...](https:\/\/www.symbolab.com\/calculators)\n- [Geometry](https:\/\/www.symbolab.com\/geometry-calculator)\n  [Pythagorean Theorem Calculator](https:\/\/www.symbolab.com\/geometry-calculator\/pythagorean-theorem-calculator) [Circle Area Calculator](https:\/\/www.symbolab.com\/geometry-calculator\/circle-area-diameter-calculator) [Isosceles Triangle Calculator](https:\/\/www.symbolab.com\/geometry-calculator\/isosceles-triangle-prove-angles-calculator) [Triangles Calculator](https:\/\/www.symbolab.com\/geometry-calculator\/triangle-perimeter-calculator) [More...](https:\/\/www.symbolab.com\/geometry-calculator#examples=true)\n- [AI Chat](https:\/\/www.symbolab.com\/ai-chat)\n- [Tools](https:\/\/www.symbolab.com\/solver\/partial-derivative-calculator)\n  [Notebook](https:\/\/www.symbolab.com\/notebook)\n  \n  [Groups](https:\/\/www.symbolab.com\/groups)\n  \n  [Cheat Sheets](https:\/\/www.symbolab.com\/cheat-sheets)\n  \n  [Worksheets](https:\/\/www.symbolab.com\/worksheets\/Pre-Algebra)\n  \n  [Study Guides](https:\/\/www.symbolab.com\/study-guides)\n  \n  [Practice](https:\/\/www.symbolab.com\/practice)\n  \n  [Verify Solution](https:\/\/www.symbolab.com\/verify)\n\n- [Upgrade](https:\/\/www.symbolab.com\/solver\/partial-derivative-calculator)\n\nClose\n\nAccept Terms\n\nGood job\\!\n\n![Practice](https:\/\/www.symbolab.com\/public\/images\/practice_icon1.png) Practice More\n\nType your Answer\n\n**Verify**\n\n|   |   |   |   |   |   |   |   |   |   |   |   |\n|---|---|---|---|---|---|---|---|---|---|---|---|\n|   |   |   |   |   |   |   |   |   |   |   |   |\n| x^2 | x^{\\\\msquare} | \\\\log\\_{\\\\msquare} | \\\\sqrt{\\\\square} | \\\\nthroot\\[\\\\msquare\\]{\\\\square} | \\\\le | \\\\ge | \\\\frac{\\\\msquare}{\\\\msquare} | \\\\cdot | \\\\div | x^{\\\\circ} | \\\\pi |\n| \\\\left(\\\\square\\\\right)^{'} | \\\\frac{d}{dx} | \\\\frac{\\\\partial}{\\\\partial x} | \\\\int | \\\\int\\_{\\\\msquare}^{\\\\msquare} | \\\\lim | \\\\sum | \\\\infty | \\\\theta | (f\\\\:\\\\circ\\\\:g) | f(x) | ![](https:\/\/www.symbolab.com\/public\/img\/pad_matrixnxn.png) |\n\n|   |   |   |   |   |   |   |   |\n|---|---|---|---|---|---|---|---|\n| ▭\\\\:\\\\longdivision{▭} | \\\\times \\\\twostack{▭}{▭} | \\+ \\\\twostack{▭}{▭} | \\- \\\\twostack{▭}{▭} | \\\\left( | \\\\right) | \\\\times | \\\\square\\\\frac{\\\\square}{\\\\square} |\n\nTake a challenge\n\nSubscribe to verify your answer\n\nSubscribe\n\nAre you sure you want to leave this Challenge? By closing this window you will lose this challenge\n\nCancel\n\nLeave\n\n- [Pre Algebra](https:\/\/www.symbolab.com\/solver\/pre-algebra-calculator)\n  [Order of Operations](https:\/\/www.symbolab.com\/solver\/order-of-operations-calculator) [Factors & Primes](https:\/\/www.symbolab.com\/solver\/factors-primes-calculator) [Fractions](https:\/\/www.symbolab.com\/solver\/fractions-calculator) [Long Arithmetic](https:\/\/www.symbolab.com\/solver\/long-arithmetic-calculator) [Decimals](https:\/\/www.symbolab.com\/solver\/decimals-calculator) [Exponents & Radicals](https:\/\/www.symbolab.com\/solver\/exponents-radicals-calculator) [Ratios & Proportions](https:\/\/www.symbolab.com\/solver\/ratios-proportions-calculator) [Percent](https:\/\/www.symbolab.com\/solver\/percent-calculator) [Modulo](https:\/\/www.symbolab.com\/solver\/modulo-calculator) [Number Line](https:\/\/www.symbolab.com\/solver\/number-line-calculator) [Expanded Form](https:\/\/www.symbolab.com\/solver\/expanded-form-calculator) [Mean, Median & Mode](https:\/\/www.symbolab.com\/solver\/mean-median-mode-calculator)\n- [Algebra](https:\/\/www.symbolab.com\/solver\/algebra-calculator)\n  [Equations](https:\/\/www.symbolab.com\/solver\/equation-calculator) [Inequalities](https:\/\/www.symbolab.com\/solver\/inequalities-calculator) [System of Equations](https:\/\/www.symbolab.com\/solver\/system-of-equations-calculator) [System of Inequalities](https:\/\/www.symbolab.com\/solver\/system-of-inequalities-calculator) [Testing Solutions](https:\/\/www.symbolab.com\/solver\/testing-solutions-calculator) [Basic Operations](https:\/\/www.symbolab.com\/solver\/basic-operations-calculator) [Algebraic Properties](https:\/\/www.symbolab.com\/solver\/algebraic-properties-calculator) [Partial Fractions](https:\/\/www.symbolab.com\/solver\/partial-fractions-calculator) [Polynomials](https:\/\/www.symbolab.com\/solver\/polynomial-calculator) [Rational Expressions](https:\/\/www.symbolab.com\/solver\/rational-expression-calculator) [Sequences](https:\/\/www.symbolab.com\/solver\/sequence-calculator) [Power Sums](https:\/\/www.symbolab.com\/solver\/power-sums-calculator) [Interval Notation](https:\/\/www.symbolab.com\/solver\/interval-notation-calculator) [Pi (Product) Notation](https:\/\/www.symbolab.com\/solver\/pi-product-notation-calculator) [Induction](https:\/\/www.symbolab.com\/solver\/induction-calculator) [Prove That](https:\/\/www.symbolab.com\/solver\/prove-that-calculator) [Logical Sets](https:\/\/www.symbolab.com\/solver\/logical-sets-calculator) [Word Problems](https:\/\/www.symbolab.com\/solver\/word-problems-calculator)\n- [Pre Calculus](https:\/\/www.symbolab.com\/solver\/pre-calculus-calculator)\n  [Equations](https:\/\/www.symbolab.com\/solver\/pre-calculus-equation-calculator) [Inequalities](https:\/\/www.symbolab.com\/solver\/pre-calculus-inequalities-calculator) [Scientific Calculator](https:\/\/www.symbolab.com\/solver\/scientific-calculator) [Scientific Notation Arithmetics](https:\/\/www.symbolab.com\/solver\/scientific-notation-arithmetics) [Complex Numbers](https:\/\/www.symbolab.com\/solver\/pre-calculus-complex-numbers-calculator) [Polar\/Cartesian](https:\/\/www.symbolab.com\/solver\/polar-cartesian-calculator) [Simultaneous Equations](https:\/\/www.symbolab.com\/solver\/simultaneous-equations-calculator) [System of Inequalities](https:\/\/www.symbolab.com\/solver\/pre-calculus-system-of-inequalities-calculator) [Polynomials](https:\/\/www.symbolab.com\/solver\/pre-calculus-polynomial-calculator) [Rationales](https:\/\/www.symbolab.com\/solver\/rationales-calculator) [Functions](https:\/\/www.symbolab.com\/solver\/pre-calculus-functions-calculator) [Arithmetic & Comp.](https:\/\/www.symbolab.com\/solver\/pre-calculus-function-arithmetic-composition-calculator) [Coordinate Geometry](https:\/\/www.symbolab.com\/solver\/pre-calculus-coordinate-geometry-calculator) [Plane Geometry](https:\/\/www.symbolab.com\/solver\/plane-geometry-calculator) [Solid Geometry](https:\/\/www.symbolab.com\/solver\/solid-geometry-calculator) [Trigonometry](https:\/\/www.symbolab.com\/solver\/pre-calculus-trigonometry-calculator)\n- [Calculus](https:\/\/www.symbolab.com\/solver\/calculus-calculator)\n  [Derivatives](https:\/\/www.symbolab.com\/solver\/derivative-calculator) [Derivative Applications](https:\/\/www.symbolab.com\/solver\/derivative-applications-calculator) [Limits](https:\/\/www.symbolab.com\/solver\/limit-calculator) [Integrals](https:\/\/www.symbolab.com\/solver\/integral-calculator) [Integral Applications](https:\/\/www.symbolab.com\/solver\/integral-applications-calculator) [Integral Approximation](https:\/\/www.symbolab.com\/solver\/integral-approximation-calculator) [Series](https:\/\/www.symbolab.com\/solver\/series-calculator) [ODE](https:\/\/www.symbolab.com\/solver\/ordinary-differential-equation-calculator) [Multivariable Calculus](https:\/\/www.symbolab.com\/solver\/multivariable-calculus-calculator) [Laplace Transform](https:\/\/www.symbolab.com\/solver\/laplace-calculator) [Taylor\/Maclaurin Series](https:\/\/www.symbolab.com\/solver\/taylor-maclaurin-series-calculator) [Fourier Series](https:\/\/www.symbolab.com\/solver\/fourier-series-calculator) [Fourier Transform](https:\/\/www.symbolab.com\/solver\/fourier-transform-calculator)\n- [Functions](https:\/\/www.symbolab.com\/solver\/functions-line-calculator)\n  [Line Equations](https:\/\/www.symbolab.com\/solver\/line-equation-calculator) [Functions](https:\/\/www.symbolab.com\/solver\/functions-calculator) [Arithmetic & Comp.](https:\/\/www.symbolab.com\/solver\/function-arithmetic-composition-calculator) [Conic Sections](https:\/\/www.symbolab.com\/solver\/conic-sections-calculator) [Transformation](https:\/\/www.symbolab.com\/solver\/function-transformation-calculator)\n- [Linear Algebra](https:\/\/www.symbolab.com\/solver\/linear-algebra-calculator)\n  [Matrices](https:\/\/www.symbolab.com\/solver\/matrix-calculator) [Vectors](https:\/\/www.symbolab.com\/solver\/vector-calculator)\n- [Trigonometry](https:\/\/www.symbolab.com\/solver\/trigonometry-calculator)\n  [Quadrant](https:\/\/www.symbolab.com\/solver\/quadrant-calculator) [Coterminal Angle](https:\/\/www.symbolab.com\/solver\/coterminal-angle-calculator) [Identities](https:\/\/www.symbolab.com\/solver\/trigonometric-identity-calculator) [Proving Identities](https:\/\/www.symbolab.com\/solver\/trigonometric-identity-proving-calculator) [Trig Equations](https:\/\/www.symbolab.com\/solver\/trigonometric-equation-calculator) [Trig Inequalities](https:\/\/www.symbolab.com\/solver\/trigonometric-inequality-calculator) [Evaluate Functions](https:\/\/www.symbolab.com\/solver\/trigonometric-function-calculator) [Simplify](https:\/\/www.symbolab.com\/solver\/trigonometric-simplification-calculator)\n- [Statistics](https:\/\/www.symbolab.com\/solver\/statistics-calculator)\n  [Mean](https:\/\/www.symbolab.com\/solver\/arithmetic-mean-calculator) [Geometric Mean](https:\/\/www.symbolab.com\/solver\/geometric-mean-calculator) [Quadratic Mean](https:\/\/www.symbolab.com\/solver\/quadratic-mean-calculator) [Average](https:\/\/www.symbolab.com\/solver\/average-calculator) [Median](https:\/\/www.symbolab.com\/solver\/median-calculator) [Mode](https:\/\/www.symbolab.com\/solver\/mode-calculator) [Order](https:\/\/www.symbolab.com\/solver\/order-calculator) [Minimum](https:\/\/www.symbolab.com\/solver\/minimum-calculator) [Maximum](https:\/\/www.symbolab.com\/solver\/maximum-calculator) [Probability](https:\/\/www.symbolab.com\/solver\/probability-calculator) [Mid-Range](https:\/\/www.symbolab.com\/solver\/mid-range-calculator) [Range](https:\/\/www.symbolab.com\/solver\/range-calculator) [Standard Deviation](https:\/\/www.symbolab.com\/solver\/standard-deviation-calculator) [Variance](https:\/\/www.symbolab.com\/solver\/variance-calculator) [Lower Quartile](https:\/\/www.symbolab.com\/solver\/lower-quartile-calculator) [Upper Quartile](https:\/\/www.symbolab.com\/solver\/upper-quartile-calculator) [Interquartile Range](https:\/\/www.symbolab.com\/solver\/interquartile-range-calculator) [Midhinge](https:\/\/www.symbolab.com\/solver\/midhinge-calculator) [Standard Normal Distribution](https:\/\/www.symbolab.com\/solver\/standard-normal-distribution-calculator)\n- [Physics](https:\/\/www.symbolab.com\/solver\/physics-calculator)\n  [Mechanics](https:\/\/www.symbolab.com\/solver\/mechanics-calculator)\n- [Chemistry](https:\/\/www.symbolab.com\/solver\/chemistry-calculator)\n  [Chemical Reactions](https:\/\/www.symbolab.com\/solver\/chemical-reaction-calculator) [Chemical Properties](https:\/\/www.symbolab.com\/solver\/chemical-properties-calculator)\n- [Finance](https:\/\/www.symbolab.com\/solver\/finance-calculator)\n  [Simple Interest](https:\/\/www.symbolab.com\/solver\/simple-interest-calculator) [Compound Interest](https:\/\/www.symbolab.com\/solver\/compound-interest-calculator) [Present Value](https:\/\/www.symbolab.com\/solver\/present-value-calculator) [Future Value](https:\/\/www.symbolab.com\/solver\/future-value-calculator)\n- [Economics](https:\/\/www.symbolab.com\/solver\/economics-calculator)\n  [Point of Diminishing Return](https:\/\/www.symbolab.com\/solver\/point-of-diminishing-return-calculator)\n- [Conversions](https:\/\/www.symbolab.com\/solver\/conversion-calculator)\n  [Currency](https:\/\/www.symbolab.com\/solver\/currency-converter) [Roman Numerals](https:\/\/www.symbolab.com\/solver\/roman-numeral-converter) [Radical to Exponent](https:\/\/www.symbolab.com\/solver\/radical-to-exponent-calculator) [Exponent to Radical](https:\/\/www.symbolab.com\/solver\/exponent-to-radical-calculator) [To Fraction](https:\/\/www.symbolab.com\/solver\/convert-to-fraction-calculator) [To Decimal](https:\/\/www.symbolab.com\/solver\/convert-to-decimal-calculator) [To Mixed Number](https:\/\/www.symbolab.com\/solver\/convert-to-mixed-number-calculator) [To Improper Fraction](https:\/\/www.symbolab.com\/solver\/convert-to-improper-fraction-calculator) [Radians to Degrees](https:\/\/www.symbolab.com\/solver\/radians-to-degrees-calculator) [Degrees to Radians](https:\/\/www.symbolab.com\/solver\/degrees-to-radians-calculator)  [Degrees Minutes Seconds](https:\/\/www.symbolab.com\/solver\/degrees-minutes-seconds-calculator) [Hexadecimal](https:\/\/www.symbolab.com\/solver\/hexadecimal-converter) [Scientific Notation](https:\/\/www.symbolab.com\/solver\/scientific-notation-converter) [Distance](https:\/\/www.symbolab.com\/solver\/distance-converter) [Weight](https:\/\/www.symbolab.com\/solver\/weight-converter) [Time](https:\/\/www.symbolab.com\/solver\/time-converter) [Volume](https:\/\/www.symbolab.com\/solver\/volume-converter)\n\n### Special Offer\n## Double the Tools. One Smart Bundle.\nto master your math and writing\\!\n\nPro\n\n- Full access to solution steps\n- Full access to AI chat\n- Practice and improve\n- Access from any device\n- No ads\n- Unlimited storage\n\n- Detect AI-generated content\n- Advanced Grammar Checker\n- AI Humanizer\n- Paraphrase in unlimited modes\n- Summarize any text\n- Much More\\!\n\n[Restrictions apply, check out our FAQs.](https:\/\/www.symbolab.com\/bundle-faq)\n\nGo back\n\nPurchase Bundle\n\n### Back to School Promotion\nAnnual\n\nAnnual - \\$\n\n%\n\nOFF\n\nAnnual plan\n\nOne time offer\n\nfor one year, then \\$\n\nGo back\n\n\\+\n\n## qb-banner-title\n###\n\n##\n###\n\n[Solutions](https:\/\/www.symbolab.com\/solver) \\> [Calculus Calculator](https:\/\/www.symbolab.com\/solver\/calculus-calculator) \\>\n\n# Partial Derivative Calculator\n\nTopic\n\n- [Pre Algebra](https:\/\/www.symbolab.com\/solver\/pre-algebra-calculator)\n- [Algebra](https:\/\/www.symbolab.com\/solver\/algebra-calculator)\n- [Pre Calculus](https:\/\/www.symbolab.com\/solver\/pre-calculus-calculator)\n- [Calculus](https:\/\/www.symbolab.com\/solver\/calculus-calculator)\n- [Derivatives]()\n  - [First Derivative](https:\/\/www.symbolab.com\/solver\/first-derivative-calculator)\n  - [WRT](https:\/\/www.symbolab.com\/solver\/derivative-with-respect-calculator)\n  - [Specify Method]()\n    - [Chain Rule](https:\/\/www.symbolab.com\/solver\/derivative-chain-rule-calculator)\n    - [Product Rule](https:\/\/www.symbolab.com\/solver\/derivative-product-rule-calculator)\n    - [Quotient Rule](https:\/\/www.symbolab.com\/solver\/derivative-quotient-rule-calculator)\n    - [Sum\/Diff Rule](https:\/\/www.symbolab.com\/solver\/derivative-sum-diff-rule-calculator)\n  - [Second Derivative](https:\/\/www.symbolab.com\/solver\/second-derivative-calculator)\n  - [Third Derivative](https:\/\/www.symbolab.com\/solver\/third-derivative-calculator)\n  - [Higher Order Derivatives](https:\/\/www.symbolab.com\/solver\/higher-order-derivative-calculator)\n  - [Derivative at a point](https:\/\/www.symbolab.com\/solver\/derivative-point-calculator)\n  - [Partial Derivative](https:\/\/www.symbolab.com\/solver\/partial-derivative-calculator)\n  - [Implicit Derivative](https:\/\/www.symbolab.com\/solver\/implicit-derivative-calculator)\n  - [Second Implicit Derivative](https:\/\/www.symbolab.com\/solver\/second-implicit-derivative-calculator)\n  - [Derivative using Definition](https:\/\/www.symbolab.com\/solver\/derivative-using-definition-calculator)\n- [Derivative Applications]()\n  - [Tangent](https:\/\/www.symbolab.com\/solver\/tangent-line-calculator)\n  - [Slope of Tangent](https:\/\/www.symbolab.com\/solver\/slope-of-tangent-calculator)\n  - [Normal](https:\/\/www.symbolab.com\/solver\/normal-line-calculator)\n  - [Curved Line Slope](https:\/\/www.symbolab.com\/solver\/curved-line-slope-calculator)\n  - [Extreme Points](https:\/\/www.symbolab.com\/solver\/calculus-function-extreme-points-calculator)\n  - [Tangent to Conic](https:\/\/www.symbolab.com\/solver\/tangent-to-conic-calculator)\n  - [Linear Approximation](https:\/\/www.symbolab.com\/solver\/linear-approximation-calculator)\n  - [Difference Quotient](https:\/\/www.symbolab.com\/solver\/difference-quotient-calculator)\n  - [Horizontal Tangent](https:\/\/www.symbolab.com\/solver\/horizontal-tangent-calculator)\n- [Limits]()\n  - [One Variable](https:\/\/www.symbolab.com\/solver\/single-var-limit-calculator)\n  - [Multi Variable Limit](https:\/\/www.symbolab.com\/solver\/multi-var-limit-calculator)\n  - [One Sided](https:\/\/www.symbolab.com\/solver\/one-sided-limit-calculator)\n  - [At Infinity](https:\/\/www.symbolab.com\/solver\/limit-infinity-calculator)\n  - [Specify Method]()\n    - [L'Hopital's Rule](https:\/\/www.symbolab.com\/solver\/limit-lhopital-calculator)\n    - [Squeeze Theorem](https:\/\/www.symbolab.com\/solver\/limit-squeeze-theorem-calculator)\n    - [Chain Rule](https:\/\/www.symbolab.com\/solver\/limit-chain-rule-calculator)\n    - [Factoring](https:\/\/www.symbolab.com\/solver\/limit-using-factoring-calculator)\n    - [Substitution](https:\/\/www.symbolab.com\/solver\/limit-substitution-calculator)\n    - [Sandwich Theorem](https:\/\/www.symbolab.com\/solver\/limit-sandwich-theorem-calculator)\n- [Integrals]()\n  - [Indefinite Integrals](https:\/\/www.symbolab.com\/solver\/indefinite-integral-calculator)\n  - [Definite Integrals](https:\/\/www.symbolab.com\/solver\/definite-integral-calculator)\n  - [Specific-Method]()\n    - [Partial Fractions](https:\/\/www.symbolab.com\/solver\/partial-fractions-integration-calculator)\n    - [U-Substitution](https:\/\/www.symbolab.com\/solver\/u-substitution-integration-calculator)\n    - [Trigonometric Substitution](https:\/\/www.symbolab.com\/solver\/trigonometric-substitution-integration-calculator)\n    - [Weierstrass Substitution](https:\/\/www.symbolab.com\/solver\/weierstrass-substitution-calculator)\n    - [By Parts](https:\/\/www.symbolab.com\/solver\/by-parts-integration-calculator)\n    - [Long Division](https:\/\/www.symbolab.com\/solver\/long-division-integration-calculator)\n  - [Improper Integrals](https:\/\/www.symbolab.com\/solver\/improper-integral-calculator)\n  - [Antiderivatives](https:\/\/www.symbolab.com\/solver\/antiderivative-calculator)\n  - [Double Integrals](https:\/\/www.symbolab.com\/solver\/double-integrals-calculator)\n  - [Triple Integrals](https:\/\/www.symbolab.com\/solver\/triple-integrals-calculator)\n  - [Multiple Integrals](https:\/\/www.symbolab.com\/solver\/multiple-integrals-calculator)\n- [Integral Applications]()\n  - [Limit of Sum](https:\/\/www.symbolab.com\/solver\/limit-of-sum-calculator)\n  - [Area under curve](https:\/\/www.symbolab.com\/solver\/area-under-curve-calculator)\n  - [Area between curves](https:\/\/www.symbolab.com\/solver\/area-between-curves-calculator)\n  - [Area under polar curve](https:\/\/www.symbolab.com\/solver\/area-under-polar-curve-calculator)\n  - [Volume of solid of revolution](https:\/\/www.symbolab.com\/solver\/volume-calculator)\n  - [Arc Length](https:\/\/www.symbolab.com\/solver\/arc-length-calculator)\n  - [Function Average](https:\/\/www.symbolab.com\/solver\/function-average-calculator)\n- [Integral Approximation]()\n  - [Riemann Sum](https:\/\/www.symbolab.com\/solver\/riemann-sum-calculator)\n  - [Trapezoidal](https:\/\/www.symbolab.com\/solver\/trapezoidal-approximation-calculator)\n  - [Simpson's Rule](https:\/\/www.symbolab.com\/solver\/simpsons-rule-calculator)\n  - [Midpoint Rule](https:\/\/www.symbolab.com\/solver\/midpoint-rule-calculator)\n- [Series]()\n  - [Convergence]()\n    - [Geometric Series Test](https:\/\/www.symbolab.com\/solver\/geometric-series-test-calculator)\n    - [Telescoping Series Test](https:\/\/www.symbolab.com\/solver\/telescoping-series-test-calculator)\n    - [Alternating Series Test](https:\/\/www.symbolab.com\/solver\/alternating-series-test-calculator)\n    - [P Series Test](https:\/\/www.symbolab.com\/solver\/p-series-test-calculator)\n    - [Divergence Test](https:\/\/www.symbolab.com\/solver\/series-divergence-test-calculator)\n    - [Ratio Test](https:\/\/www.symbolab.com\/solver\/series-ratio-test-calculator)\n    - [Root Test](https:\/\/www.symbolab.com\/solver\/series-root-test-calculator)\n    - [Comparison Test](https:\/\/www.symbolab.com\/solver\/series-comparison-test-calculator)\n    - [Limit Comparison Test](https:\/\/www.symbolab.com\/solver\/series-limit-comparison-test-calculator)\n    - [Integral Test](https:\/\/www.symbolab.com\/solver\/series-integral-test-calculator)\n  - [Absolute Convergence](https:\/\/www.symbolab.com\/solver\/series-absolute-convergence-calculator)\n  - [Power Series]()\n    - [Radius of Convergence](https:\/\/www.symbolab.com\/solver\/radius-of-convergence-calculator)\n    - [Interval of Convergence](https:\/\/www.symbolab.com\/solver\/interval-of-convergence-calculator)\n- [ODE]()\n  - [Linear First Order](https:\/\/www.symbolab.com\/solver\/linear-first-order-differential-equation-calculator)\n  - [Linear w\/constant coefficients](https:\/\/www.symbolab.com\/solver\/linear-constant-coefficients-differential-equation-calculator)\n  - [Separable](https:\/\/www.symbolab.com\/solver\/separable-differential-equation-calculator)\n  - [Bernoulli](https:\/\/www.symbolab.com\/solver\/bernoulli-differential-equation-calculator)\n  - [Exact](https:\/\/www.symbolab.com\/solver\/exact-differential-equation-calculator)\n  - [Second Order](https:\/\/www.symbolab.com\/solver\/second-order-differential-equation-calculator)\n  - [Homogenous](https:\/\/www.symbolab.com\/solver\/homogenous-differential-equation-calculator)\n  - [Non Homogenous](https:\/\/www.symbolab.com\/solver\/non-homogenous-differential-equation-calculator)\n  - [Substitution](https:\/\/www.symbolab.com\/solver\/substitution-differential-equation-calculator)\n  - [System of ODEs](https:\/\/www.symbolab.com\/solver\/system-of-differential-equations-calculator)\n  - [IVP using Laplace](https:\/\/www.symbolab.com\/solver\/ivp-laplace-calculator)\n  - [Series Solutions](https:\/\/www.symbolab.com\/solver\/ode-series-solutions-calculator)\n  - [Method of Frobenius](https:\/\/www.symbolab.com\/solver\/frobenius-method-calculator)\n  - [Gamma Function](https:\/\/www.symbolab.com\/solver\/gamma-function-calculator)\n- [Multivariable Calculus]()\n  - [Partial Derivative](https:\/\/www.symbolab.com\/solver\/multivariable-partial-derivative-calculator)\n  - [Implicit Derivative](https:\/\/www.symbolab.com\/solver\/multivariable-implicit-derivative-calculator)\n  - [Tangent to Conic](https:\/\/www.symbolab.com\/solver\/multivariable-tangent-to-conic-calculator)\n  - [Multi Variable Limit](https:\/\/www.symbolab.com\/solver\/multi-var-calculus-limit-calculator)\n  - [Multiple Integrals](https:\/\/www.symbolab.com\/solver\/multiple-integrals-multivariable-calculator)\n  - [Gradient](https:\/\/www.symbolab.com\/solver\/gradient-calculator)\n  - [Divergence](https:\/\/www.symbolab.com\/solver\/divergence-calculator)\n  - [Extreme Points](https:\/\/www.symbolab.com\/solver\/multi-var-function-extreme-points-calculator)\n- [Laplace Transform]()\n  - [Inverse](https:\/\/www.symbolab.com\/solver\/inverse-laplace-calculator)\n- [Taylor\/Maclaurin Series]()\n  - [Taylor Series](https:\/\/www.symbolab.com\/solver\/taylor-series-calculator)\n  - [Maclaurin Series](https:\/\/www.symbolab.com\/solver\/maclaurin-series-calculator)\n- [Fourier Series](https:\/\/www.symbolab.com\/solver\/fourier-series-calculator)\n- [Fourier Transform](https:\/\/www.symbolab.com\/solver\/fourier-transform-calculator)\n- [Functions](https:\/\/www.symbolab.com\/solver\/functions-line-calculator)\n- [Linear Algebra](https:\/\/www.symbolab.com\/solver\/linear-algebra-calculator)\n- [Trigonometry](https:\/\/www.symbolab.com\/solver\/trigonometry-calculator)\n- [Statistics](https:\/\/www.symbolab.com\/solver\/statistics-calculator)\n- [Physics](https:\/\/www.symbolab.com\/solver\/physics-calculator)\n- [Chemistry](https:\/\/www.symbolab.com\/solver\/chemistry-calculator)\n- [Finance](https:\/\/www.symbolab.com\/solver\/finance-calculator)\n- [Economics](https:\/\/www.symbolab.com\/solver\/economics-calculator)\n- [Conversions](https:\/\/www.symbolab.com\/solver\/conversion-calculator)\n\nAdd to Chrome\n\nGet our extension, you can capture any math problem from any website\n\nFull pad\n\n|   |   |   |   |   |   |   |   |   |   |   |   |\n|---|---|---|---|---|---|---|---|---|---|---|---|\n|   |   |   |   |   |   |   |   |   |   |   |   |\n| x^2 | x^{\\\\msquare} | \\\\log\\_{\\\\msquare} | \\\\sqrt{\\\\square} | \\\\nthroot\\[\\\\msquare\\]{\\\\square} | \\\\le | \\\\ge | \\\\frac{\\\\msquare}{\\\\msquare} | \\\\cdot | \\\\div | x^{\\\\circ} | \\\\pi |\n| \\\\left(\\\\square\\\\right)^{'} | \\\\frac{d}{dx} | \\\\frac{\\\\partial}{\\\\partial x} | \\\\int | \\\\int\\_{\\\\msquare}^{\\\\msquare} | \\\\lim | \\\\sum | \\\\infty | \\\\theta | (f\\\\:\\\\circ\\\\:g) | f(x) | ![](https:\/\/www.symbolab.com\/public\/img\/pad_matrixnxn.png) |\n\n|   |   |   |   |   |   |   |\n|---|---|---|---|---|---|---|\n| \\- \\\\twostack{▭}{▭} | \\\\lt | 7 | 8 | 9 | \\\\div | AC |\n| \\+ \\\\twostack{▭}{▭} | \\\\gt | 4 | 5 | 6 | \\\\times | \\\\square\\\\frac{\\\\square}{\\\\square} |\n| \\\\times \\\\twostack{▭}{▭} | \\\\left( | 1 | 2 | 3 | \\- | x |\n| ▭\\\\:\\\\longdivision{▭} | \\\\right) | . | 0 | \\= | \\+ | y |\n\n\\\\mathrm{implicit\\\\:derivative}    \\\\mathrm{tangent}    \\\\mathrm{volume}    \\\\mathrm{laplace}    \\\\mathrm{fourier}\n\nSee All\n\narea\n\nasymptotes\n\ncritical points\n\nderivative\n\ndomain\n\neigenvalues\n\neigenvectors\n\nexpand\n\nextreme points\n\nfactor\n\nimplicit derivative\n\ninflection points\n\nintercepts\n\ninverse\n\nlaplace\n\ninverse laplace\n\npartial fractions\n\nrange\n\nslope\n\nsimplify\n\nsolve for\n\ntangent\n\ntaylor\n\nvertex\n\ngeometric test\n\nalternating test\n\ntelescoping test\n\npseries test\n\nroot test\n\n**Go**\n\n[Steps](https:\/\/www.symbolab.com\/solver\/partial-derivative-calculator#steps) [Graph](https:\/\/www.symbolab.com\/solver\/partial-derivative-calculator#plot) [Related](https:\/\/www.symbolab.com\/solver\/partial-derivative-calculator#related_anchor) [Examples](https:\/\/www.symbolab.com\/solver\/partial-derivative-calculator#Examples)\n\nGenerated by AI\n\nAI explanations are generated using OpenAI technology. AI generated content may present inaccurate or offensive content that does not represent Symbolab's view.\n\nVerify your Answer\n\nSubscribe to verify your answer\n\nSubscribe\n\nSave to Notebook\\!\n\nSign in to save notes\n\nSign in\n\nVerify\n\nSave\n\nShow Steps\n\nHide Steps\n\n## Number Line\n![]()\n\n![](https:\/\/www.symbolab.com\/public\/img\/ajax-loader.gif)\n\nRelated\n\nPartial Derivative Examples\n\n- [\\\\frac{\\\\partial}{\\\\partial x}(\\\\sin (x^2y^2))](https:\/\/www.symbolab.com\/solver\/partial-derivative-calculator\/%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20x%7D%28%5Csin%20%28x%5E2y%5E2%29%29?or=ex)\n- [\\\\frac{\\\\partial}{\\\\partial y}(\\\\sin (x^2y^2))](https:\/\/www.symbolab.com\/solver\/partial-derivative-calculator\/%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20y%7D%28%5Csin%20%28x%5E2y%5E2%29%29?or=ex)\n- [\\\\frac{\\\\partial}{\\\\partial y\\\\partial x}(\\\\sin (x^2y^2))](https:\/\/www.symbolab.com\/solver\/partial-derivative-calculator\/%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20y%5Cpartial%20x%7D%28%5Csin%20%28x%5E2y%5E2%29%29?or=ex)\n- [\\\\frac{\\\\partial}{\\\\partial w}(te^{(\\\\frac{w}{t})})](https:\/\/www.symbolab.com\/solver\/partial-derivative-calculator\/%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20w%7D%28te%5E%7B%28%5Cfrac%7Bw%7D%7Bt%7D%29%7D%29?or=ex)\n- [\\\\frac{\\\\partial}{\\\\partial t}(te^{(\\\\frac{w}{t})})](https:\/\/www.symbolab.com\/solver\/partial-derivative-calculator\/%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20t%7D%28te%5E%7B%28%5Cfrac%7Bw%7D%7Bt%7D%29%7D%29?or=ex)\n- [\\\\frac{\\\\partial}{\\\\partial v}(\\\\sqrt{u^2+v^2})](https:\/\/www.symbolab.com\/solver\/partial-derivative-calculator\/%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20v%7D%28%5Csqrt%7Bu%5E2%2Bv%5E2%7D%29?or=ex)\n- Show More\n\n## A Step-by-Step Guide to Partial Derivatives Calculator\nMany moments in daily life hinge on more than one changing factor. The taste of homemade soup, for example, is not just about the mix of spices but also the time the pot spends simmering. A perfect photograph is shaped by both the [angle](https:\/\/www.symbolab.com\/geometry-calculator\/angles-calculator) of the light and the split-second timing of the shutter. There’s rarely only one ingredient at play. When several [variables](https:\/\/www.symbolab.com\/solver\/solve-for-equation-calculator) come together to shape an outcome, partial derivatives offer a quiet way to notice: what happens if only one piece of the puzzle moves, while everything else stands still? In this article, partial derivatives will be explored one careful step at a time—what they are, why they matter, how they show up in daily life, and how to work with them using Symbolab’s Partial Derivative Calculator.\n\n## What are Partial Derivatives\nThink of any outcome that depends on a handful of choices, not just one. In math, these relationships are described by [functions](https:\/\/www.symbolab.com\/solver\/functions-calculator) like \\$f(x, y)\\$, where both \\$x\\$ and \\$y\\$ are inputs—sometimes these are hours studied and hours slept, sometimes they’re temperature and humidity, sometimes they’re the amount of flour and the oven’s heat. Each factor quietly pushes and pulls the result in its own way.\n\nA partial derivative asks: if one variable changes—say, \\$x\\$—while \\$y\\$ stays exactly the same, how does the outcome respond? It’s like gently turning one dial on a stereo and listening for just that note in the music. Mathematically, the partial derivative with respect to \\$x\\$ is written as:\n\n\\$\\\\frac{\\\\partial f}{\\\\partial x}\\$\n\nand the same question about \\$y\\$ gives:\n\n\\$\\\\frac{\\\\partial f}{\\\\partial y}\\$\n\nNo need to let the symbols overwhelm; the heart of it is this: only one input shifts, the rest are frozen in place.\n\nFor something even more down-to-earth, think about cooking rice. The final texture, \\$T\\$, depends on both water \\$w\\$ and heat \\$h\\$—so \\$T(w, h)\\$. Suppose you want to know how the rice’s texture changes when you increase the heat while keeping the water level constant. That rate of change is captured by the partial derivative with respect to heat, written as:\n\n\\$\\\\frac{\\\\partial T}{\\\\partial h}\\$\n\nmeasures just that. How does the rice change, with water unchanged, as the flame turns up?\n\nThis idea isn’t only for kitchens or science labs; it appears wherever complex outcomes depend on several choices at once. Partial derivatives let us slow down and focus, revealing how small, single changes can shape a bigger story. They’re a first, essential step toward making sense of the whole beautiful tangle of variables that fill real life.\n\n## Understanding Multivariable Functions\nIt’s rare for one single variable to hold all the power. Most real-world outcomes—the brightness of a room, the total at a checkout counter, the time it takes to bike across town—depend on two or more factors working together. In math, multivariable functions capture these relationships.\n\nA multivariable function might look like \\$f(x, y)\\$, where both \\$x\\$ and \\$y\\$ influence the result. Think about a weather app that predicts temperature based on time of day (\\$t\\$) and cloud cover (\\$c\\$), or a bakery’s profit, \\$P(c, p)\\$, that depends on the number of cakes (\\$c\\$) and pies (\\$p\\$) sold. Each variable tugs on the outcome in its own way.\n\nFor example:\n\nIn a classroom, a student’s test score might depend on hours spent studying (\\$s\\$) and hours slept the night before (\\$h\\$): \\$S(s, h)\\$. A family’s grocery bill could be modeled as \\$B(f, v)\\$, where \\$f\\$ is pounds of fruit and \\$v\\$ is pounds of vegetables in the cart.\n\nSometimes, these functions are simple sums:\n\n\\$f(x,y)=2x+3y\\$\n\nOther times, they weave together in more interesting patterns, like\n\n\\$f(x, y) = x^2 + xy + y^2\\$\n\nor\n\n\\$f(x, y, z) = xyz\\$\n\nMultivariable functions create a kind of landscape—each combination of inputs gives a new point in that landscape. Understanding how each variable shapes the outcome, one at a time, is where partial derivatives enter the scene. But before calculating changes, it helps to pause and notice: in math as in life, the most interesting results often depend on more than one ingredient.\n\n## The Concept of Partial Derivatives\nPicture a hill stretching out in every direction. The height at any point depends on how far north or east you walk. This is a classic multivariable function—height as a function of \\$x\\$ (east-west) and \\$y\\$ (north-south): \\$h(x, y)\\$. Now, imagine standing still in the north-south direction, and taking a small step east. The question becomes: how does the height change as you move only east, with your north-south position unchanged?\n\nThat is the spirit of a partial derivative. It measures how a function’s output shifts as you change one input ever so slightly, while all other variables are frozen. In mathematical language, for a function \\$f(x, y)\\$, the partial derivative with respect to \\$x\\$ tells the rate at which \\$f\\$ increases or decreases if only \\$x\\$ is allowed to change:\n\n\\$\\\\frac{\\\\partial f}{\\\\partial x}\\$\n\nIn the world beyond math, this might look like asking: How does a plant’s growth respond to extra sunlight if water stays constant? Or, how does the comfort in a room shift if only the temperature rises, while humidity does not budge?\n\nThe core idea is focus—tuning out every other influence for a moment and seeing the effect of just one change. It’s a small, controlled experiment, repeated for each variable in turn. Partial derivatives offer a way to trace the shape of a complex system, one direction at a time, revealing how each thread is woven into the whole.\n\n## Notation and Mathematical Definition of Partial Derivatives\nAt first, the notation for partial derivatives might look unfamiliar, but each symbol carries a clear meaning. The most common is the “curly d,” written as \\$\\\\partial\\$, which signals that only one variable is changing at a time.\n\nFor a function \\$f(x, y)\\$, the partial derivative with respect to \\$x\\$ is written as:\n\n\\$\\\\frac{\\\\partial f}{\\\\partial x}\\$\n\nThis is read as “the partial derivative of \\$f\\$ with respect to \\$x\\$.” Here, \\$y\\$ is held constant—almost as if \\$y\\$ has been pressed on pause.\n\nThe process works the same for \\$y\\$:\n\n\\$\\\\frac{\\\\partial f}{\\\\partial y}\\$\n\nAnd if there are more variables, the pattern continues. For \\$f(x, y, z)\\$, you might see:\n\n\\$\\\\frac{\\\\partial f}{\\\\partial z}\\$\n\n### The Formal Definition\nMathematically, the partial derivative is defined as a limit. For \\$f(x, y)\\$, the partial derivative with respect to \\$x\\$ at a specific point \\$(a, b)\\$ is:\n\n\\$\\\\displaystyle \\\\frac{\\\\partial f}{\\\\partial x}(a,b) = \\\\lim\\_{h \\\\to 0} \\\\frac{f(a+h,,b) - f(a,,b)}{h}\\$\n\nThis formula captures what happens to \\$f\\$ as \\$x\\$ increases by a tiny amount \\$h\\$, while \\$y\\$ remains fixed at \\$b\\$.\n\nNo matter how many variables there are, the logic is always the same: only one is allowed to move, the rest are temporarily held steady. With practice, these symbols become a way to trace the changing shape of any landscape, mathematical or otherwise.\n\n## Higher-Order and Mixed Partial Derivatives\n### Higher-Order Partial Derivatives\nA higher-order partial derivative is found by taking the partial derivative of a partial derivative. For example, take a function \\$f(x, y)\\$. The first partial derivative with respect to \\$x\\$ is \\$\\\\frac{\\\\partial f}{\\\\partial x}\\$. If this is differentiated again with respect to \\$x\\$, the result is the second-order partial derivative:\n\n\\$\\\\frac{\\\\partial^2 f}{\\\\partial x^2}\\$\n\nThis measures how the rate of change with respect to \\$x\\$ changes as \\$x\\$ continues to change. Imagine climbing a hill: the second-order derivative tells if the slope is getting steeper or more gentle as you move east.\n\n### Mixed Partial Derivatives\nMixed partial derivatives measure how the rate of change with respect to one variable shifts as another variable changes. For \\$f(x, y)\\$, the mixed partial derivative with respect to \\$x\\$ and then \\$y\\$ is:\n\n\\$\\\\frac{\\\\partial^2 f}{\\\\partial y \\\\partial x}\\$\n\nThis means: first, find the partial derivative with respect to \\$x\\$, then take the partial derivative of that result with respect to \\$y\\$. In real life, this might answer questions like, “How does the effect of more study hours on a test score change as sleep hours increase?” If the function is well-behaved (meaning it is “nice enough,” with no sharp corners or breaks), the mixed partial derivatives can be taken in any order and still get the same answer:\n\n\\$\\\\frac{\\\\partial^2 f}{\\\\partial y \\\\partial x} = \\\\frac{\\\\partial^2 f}{\\\\partial x \\\\partial y}\\$\n\n## Interpreting Partial Derivatives\nInterpreting a partial derivative means understanding what the value represents in a specific context. For a function \\$f(x, y)\\$, the partial derivative \\$\\\\frac{\\\\partial f}{\\\\partial x}\\$ describes how much \\$f\\$ will increase or decrease if \\$x\\$ changes by a tiny amount, while \\$y\\$ is held constant.\n\nFor example, if \\$f(x, y)\\$ gives the amount of money earned at a lemonade stand, where \\$x\\$ is the number of lemons used and \\$y\\$ is the cups of sugar, then \\$\\\\frac{\\\\partial f}{\\\\partial x}\\$ tells how much extra money is earned for each extra lemon added—assuming the sugar stays the same. This makes it possible to see, one step at a time, how a single choice affects the result.\n\n## Real-World Applications of Partial Derivatives\nPartial derivatives might seem abstract at first glance, but their fingerprints are everywhere in daily life and across nearly every field that measures, predicts, or designs.\n\n- In economics, \\$\\\\frac{\\\\partial P}{\\\\partial n}\\$ measures how profit changes with each additional product sold, keeping advertising constant.\n- In medicine, \\$\\\\frac{\\\\partial R}{\\\\partial d}\\$ shows how a patient’s response varies with drug dosage, while sleep remains fixed.\n- In engineering, \\$\\\\frac{\\\\partial S}{\\\\partial T}\\$ reveals how material stress shifts with temperature, force held steady.\n- In weather forecasting, \\$\\\\frac{\\\\partial T}{\\\\partial h}\\$ tells how temperature changes with altitude at a given location.\n- In environmental science, \\$\\\\frac{\\\\partial C}{\\\\partial t}\\$ describes how pollution concentration changes over time at one spot.\n- In machine learning, \\$\\\\frac{\\\\partial L}{\\\\partial w}\\$ helps algorithms adjust model weights to improve prediction accuracy.\n## How to Calculate Partial Derivatives Manually\nSolving partial derivatives by hand is often about careful focus—paying attention to one variable, while treating all others as if they’re constants. Here’s how to approach it, step by step:\n\n### 1\\. Pick the Variable\nDecide which variable you’re differentiating with respect to—call it \\$x\\$. All other variables (like \\$y\\$, \\$z\\$, etc.) are held constant during this step.\n\n### 2\\. Differentiate as Usual\nApply the rules of differentiation to \\$x\\$, just as you would for a single-variable function. If you see a \\$y\\$ or \\$z\\$, treat it as if it were just a number.\n\n### 3\\. Leave Other Variables Alone\nVariables that aren’t the one you’re differentiating by don’t change. If the term doesn’t contain \\$x\\$, its [derivative](https:\/\/www.symbolab.com\/solver\/derivative-calculator) with respect to \\$x\\$ is \\$0\\$.\n\nExample\n\nSuppose \\$f(x, y) = 3x^2y + 2y^2 + 7x\\$.\n\nTo find the partial derivative with respect to \\$x\\$:\n\n- \\$3x^2y\\$ becomes \\$6xy\\$ (since \\$y\\$ is treated as constant)\n- \\$2y^2\\$ becomes \\$0\\$ (no \\$x\\$ in this term)\n- \\$7x\\$ becomes \\$7\\$\n\nSo,\n\n\\$\\\\frac{\\\\partial f}{\\\\partial x} = 6xy + 7\\$\n\nTo find the partial derivative with respect to \\$y\\$:\n\n- \\$3x^2y\\$ becomes \\$3x^2\\$ (since \\$x\\$ is treated as constant)\n- \\$2y^2\\$ becomes \\$4y\\$\n- \\$7x\\$ becomes \\$0\\$\n\nSo,\n\n\\$\\\\frac{\\\\partial f}{\\\\partial y} = 3x^2 + 4y\\$\n\nThis approach works for any function of two or more variables. Keep your focus on just one variable at a time, and the process becomes much clearer. Calculating partial derivatives manually not only builds understanding, but also strengthens confidence for more complex problems.\n\n## Common Mistakes and How to Avoid Them\nPartial derivatives are powerful, but a few classic missteps can trip up even careful students. Here are some common mistakes—plus gentle reminders to help keep your work on track:\n\n- **Forgetting to Hold Other Variables Constant:** When differentiating with respect to \\$x\\$, every other variable (like \\$y\\$ or \\$z\\$) should be treated as if it is a fixed number. Only \\$x\\$ is “alive” in that moment.\n- **Differentiating Terms Without the Chosen Variable:** Any term that does not contain the variable you are differentiating with respect to has a derivative of \\$0\\$ for that step. For example, in \\$\\\\frac{\\\\partial}{\\\\partial x}(5y^2)\\$, the answer is \\$0\\$.\n- **Mixing Up Notation:** Using \\$\\\\frac{d}{dx}\\$ instead of \\$\\\\frac{\\\\partial}{\\\\partial x}\\$, or forgetting the curly \\$\\\\partial\\$, is a small but common slip. Remember, partial derivatives always use the \\$\\\\partial\\$ symbol.\n- **Dropping Variables Too Early:** Don’t remove variables from your answer unless the differentiation process sets them to \\$0\\$. For example, \\$3x^2y\\$ differentiated with respect to \\$x\\$ becomes \\$6xy\\$, not just \\$6x\\$.\n- **Forgetting Which Variable is “In Play”:** Always double-check which variable you’re differentiating by, especially in functions with several variables.\n## How to Use the Partial Derivative Calculator on Symbolab\nWhen calculations get lengthy or you want extra reassurance, Symbolab’s Partial Derivative Calculator is a supportive tool for every learner. Here’s how to use it, step by step:\n\n### Step 1: Enter the Expression\n- Type directly using your keyboard: Enter your function, such as \\$f(x, y) = 3x^2y + 2y^2 + 7x\\$.\n- Use the math keyboard: Handy for entering square roots, fractions, [exponents](https:\/\/www.symbolab.com\/solver\/exponents-calculator), and other special math symbols.\n- Upload a photo: Snap a picture of a handwritten problem or textbook page.\n- Use the Chrome extension: Capture math directly from a webpage by highlighting and screenshotting the expression. Press the “Go” button.\n### Step 2: View Step-by-Step Breakdown\nWatch as each part of the solution is explained clearly, with the option to go through one step at a time.\n\nIf a step doesn’t make sense, or if you’re curious about a detail, use the ‘Chat with Symbo’ to ask questions or get clarifications tailored to your needs.\n\n## Conclusion\nPartial derivatives are more than a set of rules—they are a way of seeing how each piece of a bigger puzzle contributes to change. Whether solved by hand or explored with Symbolab’s calculator, each calculation reveals a little more about how the world’s variables interact. With patience and practice, these tools help make complex situations understandable, guiding thoughtful decisions in science, business, engineering, or everyday life.\n\nFrequently Asked Questions (FAQ)\n\n- How do you find the partial derivative?\n- To calculate the partial derivative of a function choose the variable with respect to which you want to take the partial derivative, and treat all the other variables as constant. Differentiate the function with respect to the chosen variable, using the rules of differentiation.\n- What are the rules of partial derivatives?\n- Partial derivatives follow the sane rules as derivatives: the sum rule, the difference rule, the product rule, the quotient rule, and the chain rule.\n- What is the sum rule of partial derivatives?\n- The sum rule of partial derivatives is a technique for calculating the partial derivative of the sum of two functions. It states that if f(x,y) and g(x,y) are both differentiable functions, then: ∂(f+g)\/∂x = ∂f\/∂x + ∂g\/∂x ∂(f+g)\/∂y = ∂f\/∂y + ∂g\/∂y\n- What is the difference rule of partial derivatives?\n- The difference rule of partial derivatives is a technique for calculating the partial derivative of the difference of two functions. It states that if f(x,y) and g(x,y) are both differentiable functions, then: ∂(f-g)\/∂x = ∂f\/∂x - ∂g\/∂x ∂(f-g)\/∂y = ∂f\/∂y - ∂g\/∂y\n- What is the product rule of partial derivatives?\n- The product rule of partial derivatives is a technique for calculating the partial derivative of the product of two functions. It states that if f(x,y) and g(x,y) are both differentiable functions, then: ∂(fg)\/∂x = f∂g\/∂x + g∂f\/∂x ∂(fg)\/∂y = f∂g\/∂y + g∂f\/∂y\n- What is the quotient rule of partial derivatives?\n- The quotient rule of partial derivatives is a technique for calculating the partial derivative of the quotient of two functions. It states that if f(x,y) and g(x,y) are both differentiable functions and g(x,y) is not equal to 0, then: ∂(f\/g)\/∂x = (∂f\/∂xg - f∂g\/∂x)\/g^2 ∂(f\/g)\/∂y = (∂f\/∂yg - f∂g\/∂y)\/g^2\n- What is the chain rule of partial derivatives?\n- The chain rule of partial derivatives is a technique for calculating the partial derivative of a composite function. It states that if f(x,y) and g(x,y) are both differentiable functions, and y is a function of x (i.e. y = h(x)), then: ∂f\/∂x = ∂f\/∂y \\* ∂y\/∂x\n- What is the partial derivative of a function?\n- The partial derivative of a function is a way of measuring how much the function changes when you change one of its variables, while holding the other variables constant.\n\npartial-derivative-calculator\n\nen\n\nRelated Symbolab blog posts\n\n[High School Math Solutions – Derivative Calculator, the Basics Differentiation is a method to calculate the rate of change (or the slope at a point on the graph); we will not...](https:\/\/blog.symbolab.com\/2014\/01\/high-school-math-solutions-derivative.html)\n\nPopular topics\n\n[scientific calculator](https:\/\/www.symbolab.com\/solver\/scientific-calculator) [inverse calculator](https:\/\/www.symbolab.com\/solver\/function-inverse-calculator) [simplify calculator](https:\/\/www.symbolab.com\/solver\/simplify-calculator) [distance calculator](https:\/\/www.symbolab.com\/solver\/distance-calculator) [fractions calculator](https:\/\/www.symbolab.com\/solver\/fractions-calculator) [interval notation calculator](https:\/\/www.symbolab.com\/solver\/interval-notation-calculator) [cross product calculator](https:\/\/www.symbolab.com\/solver\/vector-cross-product-calculator) [probability calculator](https:\/\/www.symbolab.com\/solver\/probability-calculator) [derivative calculator](https:\/\/www.symbolab.com\/solver\/derivative-calculator) [series calculator](https:\/\/www.symbolab.com\/solver\/series-calculator) [ratios calculator](https:\/\/www.symbolab.com\/solver\/ratios-calculator) [statistics calculator](https:\/\/www.symbolab.com\/solver\/statistics-calculator) [integral calculator](https:\/\/www.symbolab.com\/solver\/integral-calculator) [inverse laplace transform calculator](https:\/\/www.symbolab.com\/solver\/inverse-laplace-calculator) [rounding calculator](https:\/\/www.symbolab.com\/solver\/rounding-calculator) [gcf calculator](https:\/\/www.symbolab.com\/solver\/gcf-calculator) [algebra calculator](https:\/\/www.symbolab.com\/solver\/algebra-calculator) [tangent line calculator](https:\/\/www.symbolab.com\/solver\/tangent-line-calculator) [trigonometry calculator](https:\/\/www.symbolab.com\/solver\/trigonometry-calculator) [log calculator](https:\/\/www.symbolab.com\/solver\/logarithmic-equation-calculator) [standard deviation calculator](https:\/\/www.symbolab.com\/solver\/standard-deviation-calculator) [linear equation calculator](https:\/\/www.symbolab.com\/solver\/linear-equation-calculator) [antiderivative calculator](https:\/\/www.symbolab.com\/solver\/antiderivative-calculator) [laplace transform calculator](https:\/\/www.symbolab.com\/solver\/laplace-calculator) [quadratic equation calculator](https:\/\/www.symbolab.com\/solver\/quadratic-equation-calculator) [domain calculator](https:\/\/www.symbolab.com\/solver\/function-domain-calculator) [decimals calculator](https:\/\/www.symbolab.com\/solver\/decimals-calculator) [limit calculator](https:\/\/www.symbolab.com\/solver\/limit-calculator) [equation solver](https:\/\/www.symbolab.com\/solver\/equation-calculator) [definite integral calculator](https:\/\/www.symbolab.com\/solver\/definite-integral-calculator) [matrix inverse calculator](https:\/\/www.symbolab.com\/solver\/matrix-inverse-calculator) [matrix calculator](https:\/\/www.symbolab.com\/solver\/matrix-calculator) [system of equations calculator](https:\/\/www.symbolab.com\/solver\/system-of-equations-calculator) [calculus calculator](https:\/\/www.symbolab.com\/solver\/calculus-calculator) [slope calculator](https:\/\/www.symbolab.com\/solver\/slope-calculator) [long division calculator](https:\/\/www.symbolab.com\/solver\/polynomial-long-division-calculator) [factors calculator](https:\/\/www.symbolab.com\/solver\/factor-calculator) [polynomial calculator](https:\/\/www.symbolab.com\/solver\/polynomial-equation-calculator) [square root calculator](https:\/\/www.symbolab.com\/solver\/square-root-calculator) [implicit differentiation calculator](https:\/\/www.symbolab.com\/solver\/implicit-derivative-calculator) [word problem solver](https:\/\/www.symbolab.com\/solver\/word-problems-calculator) [differential equation calculator](https:\/\/www.symbolab.com\/solver\/ordinary-differential-equation-calculator) [average calculator](https:\/\/www.symbolab.com\/calculator\/statistics\/average) [synthetic division calculator](https:\/\/www.symbolab.com\/solver\/synthetic-division-calculator)\n\nChat with Symbo AI\n\nAI may present inaccurate or offensive content that does not represent Symbolab's views.\n\nDo not enter any personal information\n\nEnter a problem\n\n[Cooking Calculators](https:\/\/www.symbolab.com\/calculator\/cooking)\n\n[Cooking Measurement Converter](https:\/\/www.symbolab.com\/calculator\/cooking\/cooking-measurement) [Cooking Ingredient Converter](https:\/\/www.symbolab.com\/calculator\/cooking\/cooking-ingredient) [Cake Pan Converter](https:\/\/www.symbolab.com\/calculator\/cooking\/cake-pan) [More calculators](https:\/\/www.symbolab.com\/calculator\/cooking)\n\n[Fitness Calculators](https:\/\/www.symbolab.com\/calculator\/fitness)\n\n[BMI Calculator](https:\/\/www.symbolab.com\/calculator\/fitness\/bmi) [Calorie Calculator](https:\/\/www.symbolab.com\/calculator\/fitness\/calorie) [BMR Calculator](https:\/\/www.symbolab.com\/calculator\/fitness\/bmr) [More calculators](https:\/\/www.symbolab.com\/calculator\/fitness)\n\nSave to Notebook\\!\n\nSign in\n\nNotebook\n\n[View Full Notebook](https:\/\/www.symbolab.com\/notebook)\n\nStudy Tools [AI Math Solver](https:\/\/www.symbolab.com\/) [AI Chat](https:\/\/www.symbolab.com\/ai-chat) [Popular Problems](https:\/\/www.symbolab.com\/popular-algebra) [Worksheets](https:\/\/www.symbolab.com\/worksheets\/Pre-Algebra) [Study Guides](https:\/\/www.symbolab.com\/study-guides) [Practice](https:\/\/www.symbolab.com\/practice) [Cheat Sheets](https:\/\/www.symbolab.com\/cheat-sheets) [Calculators](https:\/\/www.symbolab.com\/calculators) [Graphing Calculator](https:\/\/www.symbolab.com\/graphing-calculator) [Geometry Calculator](https:\/\/www.symbolab.com\/geometry-calculator) [Verify Solution](https:\/\/www.symbolab.com\/verify)\n\nApps [Symbolab App (Android)](https:\/\/play.google.com\/store\/apps\/details?id=com.devsense.symbolab) [Graphing Calculator (Android)](https:\/\/play.google.com\/store\/apps\/details?id=com.symbolab.graphingcalculator) [Practice (Android)](https:\/\/play.google.com\/store\/apps\/details?id=com.symbolab.practice) [Symbolab App (iOS)](https:\/\/apps.apple.com\/us\/app\/id876942533) [Graphing Calculator (iOS)](https:\/\/apps.apple.com\/us\/app\/id1435182798) [Practice (iOS)](https:\/\/apps.apple.com\/us\/app\/id1469186281) [Chrome Extension](https:\/\/chromewebstore.google.com\/detail\/mgfcnnpakbfcefgphceolkjldjfpieff)\n\nCompany [About Symbolab](https:\/\/www.symbolab.com\/about) [Blog](https:\/\/blog.symbolab.com\/) [Help](https:\/\/www.symbolab.com\/help) [Contact Us](https:\/\/www.symbolab.com\/solver\/partial-derivative-calculator)\n\nLegal [Privacy](https:\/\/www.symbolab.com\/privacy-policy) [Service Terms](https:\/\/www.symbolab.com\/terms-of-use) [Cookie Policy](https:\/\/www.symbolab.com\/cookie-policy) Cookie Settings Copyright, Community Guidelines, DSA & other Legal Resources [Learneo Legal Center](https:\/\/learneo.com\/legal)\n\n[Feedback](https:\/\/www.symbolab.com\/solver\/partial-derivative-calculator)\n\nSocial Media\n\nSymbolab, a Learneo, Inc. business\n\n© Learneo, Inc. 2024\n***\n(optional)\n\n(optional)\n\nPlease add a message.\n\nMessage received. Thanks for the feedback.\n\nCancel **Send**\n\n![](https:\/\/www.symbolab.com\/public\/img\/ajax-loader.gif)\n\n![]()","attrs_readable_markdown":"## A Step-by-Step Guide to Partial Derivatives Calculator\nMany moments in daily life hinge on more than one changing factor. The taste of homemade soup, for example, is not just about the mix of spices but also the time the pot spends simmering. A perfect photograph is shaped by both the [angle](https:\/\/www.symbolab.com\/geometry-calculator\/angles-calculator) of the light and the split-second timing of the shutter. There’s rarely only one ingredient at play. When several [variables](https:\/\/www.symbolab.com\/solver\/solve-for-equation-calculator) come together to shape an outcome, partial derivatives offer a quiet way to notice: what happens if only one piece of the puzzle moves, while everything else stands still? In this article, partial derivatives will be explored one careful step at a time—what they are, why they matter, how they show up in daily life, and how to work with them using Symbolab’s Partial Derivative Calculator.\n\n## What are Partial Derivatives\nThink of any outcome that depends on a handful of choices, not just one. In math, these relationships are described by [functions](https:\/\/www.symbolab.com\/solver\/functions-calculator) like \\$f(x, y)\\$, where both \\$x\\$ and \\$y\\$ are inputs—sometimes these are hours studied and hours slept, sometimes they’re temperature and humidity, sometimes they’re the amount of flour and the oven’s heat. Each factor quietly pushes and pulls the result in its own way.\n\nA partial derivative asks: if one variable changes—say, \\$x\\$—while \\$y\\$ stays exactly the same, how does the outcome respond? It’s like gently turning one dial on a stereo and listening for just that note in the music. Mathematically, the partial derivative with respect to \\$x\\$ is written as:\n\n\\$\\\\frac{\\\\partial f}{\\\\partial x}\\$\n\nand the same question about \\$y\\$ gives:\n\n\\$\\\\frac{\\\\partial f}{\\\\partial y}\\$\n\nNo need to let the symbols overwhelm; the heart of it is this: only one input shifts, the rest are frozen in place.\n\nFor something even more down-to-earth, think about cooking rice. The final texture, \\$T\\$, depends on both water \\$w\\$ and heat \\$h\\$—so \\$T(w, h)\\$. Suppose you want to know how the rice’s texture changes when you increase the heat while keeping the water level constant. That rate of change is captured by the partial derivative with respect to heat, written as:\n\n\\$\\\\frac{\\\\partial T}{\\\\partial h}\\$\n\nmeasures just that. How does the rice change, with water unchanged, as the flame turns up?\n\nThis idea isn’t only for kitchens or science labs; it appears wherever complex outcomes depend on several choices at once. Partial derivatives let us slow down and focus, revealing how small, single changes can shape a bigger story. They’re a first, essential step toward making sense of the whole beautiful tangle of variables that fill real life.\n\n## Understanding Multivariable Functions\nIt’s rare for one single variable to hold all the power. Most real-world outcomes—the brightness of a room, the total at a checkout counter, the time it takes to bike across town—depend on two or more factors working together. In math, multivariable functions capture these relationships.\n\nA multivariable function might look like \\$f(x, y)\\$, where both \\$x\\$ and \\$y\\$ influence the result. Think about a weather app that predicts temperature based on time of day (\\$t\\$) and cloud cover (\\$c\\$), or a bakery’s profit, \\$P(c, p)\\$, that depends on the number of cakes (\\$c\\$) and pies (\\$p\\$) sold. Each variable tugs on the outcome in its own way.\n\nFor example:\n\nIn a classroom, a student’s test score might depend on hours spent studying (\\$s\\$) and hours slept the night before (\\$h\\$): \\$S(s, h)\\$. A family’s grocery bill could be modeled as \\$B(f, v)\\$, where \\$f\\$ is pounds of fruit and \\$v\\$ is pounds of vegetables in the cart.\n\nSometimes, these functions are simple sums:\n\n\\$f(x,y)=2x+3y\\$\n\nOther times, they weave together in more interesting patterns, like\n\n\\$f(x, y) = x^2 + xy + y^2\\$\n\nor\n\n\\$f(x, y, z) = xyz\\$\n\nMultivariable functions create a kind of landscape—each combination of inputs gives a new point in that landscape. Understanding how each variable shapes the outcome, one at a time, is where partial derivatives enter the scene. But before calculating changes, it helps to pause and notice: in math as in life, the most interesting results often depend on more than one ingredient.\n\n## The Concept of Partial Derivatives\nPicture a hill stretching out in every direction. The height at any point depends on how far north or east you walk. This is a classic multivariable function—height as a function of \\$x\\$ (east-west) and \\$y\\$ (north-south): \\$h(x, y)\\$. Now, imagine standing still in the north-south direction, and taking a small step east. The question becomes: how does the height change as you move only east, with your north-south position unchanged?\n\nThat is the spirit of a partial derivative. It measures how a function’s output shifts as you change one input ever so slightly, while all other variables are frozen. In mathematical language, for a function \\$f(x, y)\\$, the partial derivative with respect to \\$x\\$ tells the rate at which \\$f\\$ increases or decreases if only \\$x\\$ is allowed to change:\n\n\\$\\\\frac{\\\\partial f}{\\\\partial x}\\$\n\nIn the world beyond math, this might look like asking: How does a plant’s growth respond to extra sunlight if water stays constant? Or, how does the comfort in a room shift if only the temperature rises, while humidity does not budge?\n\nThe core idea is focus—tuning out every other influence for a moment and seeing the effect of just one change. It’s a small, controlled experiment, repeated for each variable in turn. Partial derivatives offer a way to trace the shape of a complex system, one direction at a time, revealing how each thread is woven into the whole.\n\n## Notation and Mathematical Definition of Partial Derivatives\nAt first, the notation for partial derivatives might look unfamiliar, but each symbol carries a clear meaning. The most common is the “curly d,” written as \\$\\\\partial\\$, which signals that only one variable is changing at a time.\n\nFor a function \\$f(x, y)\\$, the partial derivative with respect to \\$x\\$ is written as:\n\n\\$\\\\frac{\\\\partial f}{\\\\partial x}\\$\n\nThis is read as “the partial derivative of \\$f\\$ with respect to \\$x\\$.” Here, \\$y\\$ is held constant—almost as if \\$y\\$ has been pressed on pause.\n\nThe process works the same for \\$y\\$:\n\n\\$\\\\frac{\\\\partial f}{\\\\partial y}\\$\n\nAnd if there are more variables, the pattern continues. For \\$f(x, y, z)\\$, you might see:\n\n\\$\\\\frac{\\\\partial f}{\\\\partial z}\\$\n\n### The Formal Definition\nMathematically, the partial derivative is defined as a limit. For \\$f(x, y)\\$, the partial derivative with respect to \\$x\\$ at a specific point \\$(a, b)\\$ is:\n\n\\$\\\\displaystyle \\\\frac{\\\\partial f}{\\\\partial x}(a,b) = \\\\lim\\_{h \\\\to 0} \\\\frac{f(a+h,,b) - f(a,,b)}{h}\\$\n\nThis formula captures what happens to \\$f\\$ as \\$x\\$ increases by a tiny amount \\$h\\$, while \\$y\\$ remains fixed at \\$b\\$.\n\nNo matter how many variables there are, the logic is always the same: only one is allowed to move, the rest are temporarily held steady. With practice, these symbols become a way to trace the changing shape of any landscape, mathematical or otherwise.\n\n## Higher-Order and Mixed Partial Derivatives\n### Higher-Order Partial Derivatives\nA higher-order partial derivative is found by taking the partial derivative of a partial derivative. For example, take a function \\$f(x, y)\\$. The first partial derivative with respect to \\$x\\$ is \\$\\\\frac{\\\\partial f}{\\\\partial x}\\$. If this is differentiated again with respect to \\$x\\$, the result is the second-order partial derivative:\n\n\\$\\\\frac{\\\\partial^2 f}{\\\\partial x^2}\\$\n\nThis measures how the rate of change with respect to \\$x\\$ changes as \\$x\\$ continues to change. Imagine climbing a hill: the second-order derivative tells if the slope is getting steeper or more gentle as you move east.\n\n### Mixed Partial Derivatives\nMixed partial derivatives measure how the rate of change with respect to one variable shifts as another variable changes. For \\$f(x, y)\\$, the mixed partial derivative with respect to \\$x\\$ and then \\$y\\$ is:\n\n\\$\\\\frac{\\\\partial^2 f}{\\\\partial y \\\\partial x}\\$\n\nThis means: first, find the partial derivative with respect to \\$x\\$, then take the partial derivative of that result with respect to \\$y\\$. In real life, this might answer questions like, “How does the effect of more study hours on a test score change as sleep hours increase?” If the function is well-behaved (meaning it is “nice enough,” with no sharp corners or breaks), the mixed partial derivatives can be taken in any order and still get the same answer:\n\n\\$\\\\frac{\\\\partial^2 f}{\\\\partial y \\\\partial x} = \\\\frac{\\\\partial^2 f}{\\\\partial x \\\\partial y}\\$\n\n## Interpreting Partial Derivatives\nInterpreting a partial derivative means understanding what the value represents in a specific context. For a function \\$f(x, y)\\$, the partial derivative \\$\\\\frac{\\\\partial f}{\\\\partial x}\\$ describes how much \\$f\\$ will increase or decrease if \\$x\\$ changes by a tiny amount, while \\$y\\$ is held constant.\n\nFor example, if \\$f(x, y)\\$ gives the amount of money earned at a lemonade stand, where \\$x\\$ is the number of lemons used and \\$y\\$ is the cups of sugar, then \\$\\\\frac{\\\\partial f}{\\\\partial x}\\$ tells how much extra money is earned for each extra lemon added—assuming the sugar stays the same. This makes it possible to see, one step at a time, how a single choice affects the result.\n\n## Real-World Applications of Partial Derivatives\nPartial derivatives might seem abstract at first glance, but their fingerprints are everywhere in daily life and across nearly every field that measures, predicts, or designs.\n\n- In economics, \\$\\\\frac{\\\\partial P}{\\\\partial n}\\$ measures how profit changes with each additional product sold, keeping advertising constant.\n- In medicine, \\$\\\\frac{\\\\partial R}{\\\\partial d}\\$ shows how a patient’s response varies with drug dosage, while sleep remains fixed.\n- In engineering, \\$\\\\frac{\\\\partial S}{\\\\partial T}\\$ reveals how material stress shifts with temperature, force held steady.\n- In weather forecasting, \\$\\\\frac{\\\\partial T}{\\\\partial h}\\$ tells how temperature changes with altitude at a given location.\n- In environmental science, \\$\\\\frac{\\\\partial C}{\\\\partial t}\\$ describes how pollution concentration changes over time at one spot.\n- In machine learning, \\$\\\\frac{\\\\partial L}{\\\\partial w}\\$ helps algorithms adjust model weights to improve prediction accuracy.\n## How to Calculate Partial Derivatives Manually\nSolving partial derivatives by hand is often about careful focus—paying attention to one variable, while treating all others as if they’re constants. Here’s how to approach it, step by step:\n\n### 1\\. Pick the Variable\nDecide which variable you’re differentiating with respect to—call it \\$x\\$. All other variables (like \\$y\\$, \\$z\\$, etc.) are held constant during this step.\n\n### 2\\. Differentiate as Usual\nApply the rules of differentiation to \\$x\\$, just as you would for a single-variable function. If you see a \\$y\\$ or \\$z\\$, treat it as if it were just a number.\n\n### 3\\. Leave Other Variables Alone\nVariables that aren’t the one you’re differentiating by don’t change. If the term doesn’t contain \\$x\\$, its [derivative](https:\/\/www.symbolab.com\/solver\/derivative-calculator) with respect to \\$x\\$ is \\$0\\$.\n\nExample\n\nSuppose \\$f(x, y) = 3x^2y + 2y^2 + 7x\\$.\n\nTo find the partial derivative with respect to \\$x\\$:\n\n- \\$3x^2y\\$ becomes \\$6xy\\$ (since \\$y\\$ is treated as constant)\n- \\$2y^2\\$ becomes \\$0\\$ (no \\$x\\$ in this term)\n- \\$7x\\$ becomes \\$7\\$\n\nSo,\n\n\\$\\\\frac{\\\\partial f}{\\\\partial x} = 6xy + 7\\$\n\nTo find the partial derivative with respect to \\$y\\$:\n\n- \\$3x^2y\\$ becomes \\$3x^2\\$ (since \\$x\\$ is treated as constant)\n- \\$2y^2\\$ becomes \\$4y\\$\n- \\$7x\\$ becomes \\$0\\$\n\nSo,\n\n\\$\\\\frac{\\\\partial f}{\\\\partial y} = 3x^2 + 4y\\$\n\nThis approach works for any function of two or more variables. Keep your focus on just one variable at a time, and the process becomes much clearer. Calculating partial derivatives manually not only builds understanding, but also strengthens confidence for more complex problems.\n\n## Common Mistakes and How to Avoid Them\nPartial derivatives are powerful, but a few classic missteps can trip up even careful students. Here are some common mistakes—plus gentle reminders to help keep your work on track:\n\n- **Forgetting to Hold Other Variables Constant:** When differentiating with respect to \\$x\\$, every other variable (like \\$y\\$ or \\$z\\$) should be treated as if it is a fixed number. Only \\$x\\$ is “alive” in that moment.\n- **Differentiating Terms Without the Chosen Variable:** Any term that does not contain the variable you are differentiating with respect to has a derivative of \\$0\\$ for that step. For example, in \\$\\\\frac{\\\\partial}{\\\\partial x}(5y^2)\\$, the answer is \\$0\\$.\n- **Mixing Up Notation:** Using \\$\\\\frac{d}{dx}\\$ instead of \\$\\\\frac{\\\\partial}{\\\\partial x}\\$, or forgetting the curly \\$\\\\partial\\$, is a small but common slip. Remember, partial derivatives always use the \\$\\\\partial\\$ symbol.\n- **Dropping Variables Too Early:** Don’t remove variables from your answer unless the differentiation process sets them to \\$0\\$. For example, \\$3x^2y\\$ differentiated with respect to \\$x\\$ becomes \\$6xy\\$, not just \\$6x\\$.\n- **Forgetting Which Variable is “In Play”:** Always double-check which variable you’re differentiating by, especially in functions with several variables.\n## How to Use the Partial Derivative Calculator on Symbolab\nWhen calculations get lengthy or you want extra reassurance, Symbolab’s Partial Derivative Calculator is a supportive tool for every learner. Here’s how to use it, step by step:\n\n### Step 1: Enter the Expression\n- Type directly using your keyboard: Enter your function, such as \\$f(x, y) = 3x^2y + 2y^2 + 7x\\$.\n- Use the math keyboard: Handy for entering square roots, fractions, [exponents](https:\/\/www.symbolab.com\/solver\/exponents-calculator), and other special math symbols.\n- Upload a photo: Snap a picture of a handwritten problem or textbook page.\n- Use the Chrome extension: Capture math directly from a webpage by highlighting and screenshotting the expression. Press the “Go” button.\n### Step 2: View Step-by-Step Breakdown\nWatch as each part of the solution is explained clearly, with the option to go through one step at a time.\n\nIf a step doesn’t make sense, or if you’re curious about a detail, use the ‘Chat with Symbo’ to ask questions or get clarifications tailored to your needs.\n\n## Conclusion\nPartial derivatives are more than a set of rules—they are a way of seeing how each piece of a bigger puzzle contributes to change. Whether solved by hand or explored with Symbolab’s calculator, each calculation reveals a little more about how the world’s variables interact. With patience and practice, these tools help make complex situations understandable, guiding thoughtful decisions in science, business, engineering, or everyday life.\n\nFrequently Asked Questions (FAQ)\n\n- How do you find the partial derivative?\n- To calculate the partial derivative of a function choose the variable with respect to which you want to take the partial derivative, and treat all the other variables as constant. Differentiate the function with respect to the chosen variable, using the rules of differentiation.\n- What are the rules of partial derivatives?\n- Partial derivatives follow the sane rules as derivatives: the sum rule, the difference rule, the product rule, the quotient rule, and the chain rule.\n- What is the sum rule of partial derivatives?\n- The sum rule of partial derivatives is a technique for calculating the partial derivative of the sum of two functions. It states that if f(x,y) and g(x,y) are both differentiable functions, then: ∂(f+g)\/∂x = ∂f\/∂x + ∂g\/∂x ∂(f+g)\/∂y = ∂f\/∂y + ∂g\/∂y\n- What is the difference rule of partial derivatives?\n- The difference rule of partial derivatives is a technique for calculating the partial derivative of the difference of two functions. It states that if f(x,y) and g(x,y) are both differentiable functions, then: ∂(f-g)\/∂x = ∂f\/∂x - ∂g\/∂x ∂(f-g)\/∂y = ∂f\/∂y - ∂g\/∂y\n- What is the product rule of partial derivatives?\n- The product rule of partial derivatives is a technique for calculating the partial derivative of the product of two functions. It states that if f(x,y) and g(x,y) are both differentiable functions, then: ∂(fg)\/∂x = f∂g\/∂x + g∂f\/∂x ∂(fg)\/∂y = f∂g\/∂y + g∂f\/∂y\n- What is the quotient rule of partial derivatives?\n- The quotient rule of partial derivatives is a technique for calculating the partial derivative of the quotient of two functions. It states that if f(x,y) and g(x,y) are both differentiable functions and g(x,y) is not equal to 0, then: ∂(f\/g)\/∂x = (∂f\/∂xg - f∂g\/∂x)\/g^2 ∂(f\/g)\/∂y = (∂f\/∂yg - f∂g\/∂y)\/g^2\n- What is the chain rule of partial derivatives?\n- The chain rule of partial derivatives is a technique for calculating the partial derivative of a composite function. It states that if f(x,y) and g(x,y) are both differentiable functions, and y is a function of x (i.e. y = h(x)), then: ∂f\/∂x = ∂f\/∂y \\* ∂y\/∂x\n- What is the partial derivative of a function?\n- The partial derivative of a function is a way of measuring how much the function changes when you change one of its variables, while holding the other variables constant.","meta_canonical":null,"ml_categories_json":"{\"\/Science\":820,\"\/Science\/Mathematics\":815,\"\/Science\/Mathematics\/Other\":690,\"\/Computers_and_Electronics\":249,\"\/Computers_and_Electronics\/Software\":182,\"\/Computers_and_Electronics\/Software\/Educational_Software\":139}","ml_types_json":"{\"\/Article\":610,\"\/Article\/Tutorial_or_Guide\":576}","ml_intent_types_json":"{\"Informational\":986,\"Commercial\":116}","meta_language":"en","attrs_author":null,"attrs_publish_time":0,"attrs_original_publish_time":1421694210,"attrs_is_republished":0,"attrs_nr_words":"3933","attrs_boilerpipe_nr_words":"2669","body_ext_links_number":17,"body_int_links_number":324,"meta_nofollow":0,"meta_noarchive":0,"props_was_rendered":0,"src_redirect":"","download_time_msec":105,"download_ttfb_msec":93,"download_size":177754}

3. Robots.txt Check

Query:

Response:

4. Spam/Ban Check

Query:

Response:

5. Seen Status Check

ℹ️ Skipped - page is already crawled

📄

INDEXABLE

✅

CRAWLED

25 days ago

🤖

ROBOTS ALLOWED

Page Info Filters

Filter	Status	Condition	Details
HTTP status	PASS	`download_http_code = 200`	HTTP 200
Age cutoff	PASS	`download_stamp > now() - 6 MONTH`	0.8 months ago
History drop	PASS	`isNull(history_drop_reason)`	No drop reason
Spam/ban	PASS	`fh_dont_index != 1 AND ml_spam_score = 0`	ml_spam_score=0
Canonical	PASS	`meta_canonical IS NULL OR = '' OR = src_unparsed`	Not set

Page Details

Property

Value

URL

https://www.symbolab.com/solver/partial-derivative-calculator

Last Crawled

2026-04-02 04:05:28 (25 days ago)

First Indexed

2015-01-19 19:03:30 (11 years ago)

HTTP Status Code

200

Content

Meta Title

Partial Derivative Calculator

Meta Description

Free partial derivative calculator - partial differentiation solver step-by-step

Meta Canonical

null

Boilerpipe Text

A Step-by-Step Guide to Partial Derivatives Calculator Many moments in daily life hinge on more than one changing factor. The taste of homemade soup, for example, is not just about the mix of spices but also the time the pot spends simmering. A perfect photograph is shaped by both the angle of the light and the split-second timing of the shutter. There’s rarely only one ingredient at play. When several variables come together to shape an outcome, partial derivatives offer a quiet way to notice: what happens if only one piece of the puzzle moves, while everything else stands still? In this article, partial derivatives will be explored one careful step at a time—what they are, why they matter, how they show up in daily life, and how to work with them using Symbolab’s Partial Derivative Calculator. What are Partial Derivatives Think of any outcome that depends on a handful of choices, not just one. In math, these relationships are described by functions like $f(x, y)$, where both $x$ and $y$ are inputs—sometimes these are hours studied and hours slept, sometimes they’re temperature and humidity, sometimes they’re the amount of flour and the oven’s heat. Each factor quietly pushes and pulls the result in its own way. A partial derivative asks: if one variable changes—say, $x$—while $y$ stays exactly the same, how does the outcome respond? It’s like gently turning one dial on a stereo and listening for just that note in the music. Mathematically, the partial derivative with respect to $x$ is written as: $\frac{\partial f}{\partial x}$ and the same question about $y$ gives: $\frac{\partial f}{\partial y}$ No need to let the symbols overwhelm; the heart of it is this: only one input shifts, the rest are frozen in place. For something even more down-to-earth, think about cooking rice. The final texture, $T$, depends on both water $w$ and heat $h$—so $T(w, h)$. Suppose you want to know how the rice’s texture changes when you increase the heat while keeping the water level constant. That rate of change is captured by the partial derivative with respect to heat, written as: $\frac{\partial T}{\partial h}$ measures just that. How does the rice change, with water unchanged, as the flame turns up? This idea isn’t only for kitchens or science labs; it appears wherever complex outcomes depend on several choices at once. Partial derivatives let us slow down and focus, revealing how small, single changes can shape a bigger story. They’re a first, essential step toward making sense of the whole beautiful tangle of variables that fill real life. Understanding Multivariable Functions It’s rare for one single variable to hold all the power. Most real-world outcomes—the brightness of a room, the total at a checkout counter, the time it takes to bike across town—depend on two or more factors working together. In math, multivariable functions capture these relationships. A multivariable function might look like $f(x, y)$, where both $x$ and $y$ influence the result. Think about a weather app that predicts temperature based on time of day ($t$) and cloud cover ($c$), or a bakery’s profit, $P(c, p)$, that depends on the number of cakes ($c$) and pies ($p$) sold. Each variable tugs on the outcome in its own way. For example: In a classroom, a student’s test score might depend on hours spent studying ($s$) and hours slept the night before ($h$): $S(s, h)$. A family’s grocery bill could be modeled as $B(f, v)$, where $f$ is pounds of fruit and $v$ is pounds of vegetables in the cart. Sometimes, these functions are simple sums: $f(x,y)=2x+3y$ Other times, they weave together in more interesting patterns, like $f(x, y) = x^2 + xy + y^2$ or $f(x, y, z) = xyz$ Multivariable functions create a kind of landscape—each combination of inputs gives a new point in that landscape. Understanding how each variable shapes the outcome, one at a time, is where partial derivatives enter the scene. But before calculating changes, it helps to pause and notice: in math as in life, the most interesting results often depend on more than one ingredient. The Concept of Partial Derivatives Picture a hill stretching out in every direction. The height at any point depends on how far north or east you walk. This is a classic multivariable function—height as a function of $x$ (east-west) and $y$ (north-south): $h(x, y)$. Now, imagine standing still in the north-south direction, and taking a small step east. The question becomes: how does the height change as you move only east, with your north-south position unchanged? That is the spirit of a partial derivative. It measures how a function’s output shifts as you change one input ever so slightly, while all other variables are frozen. In mathematical language, for a function $f(x, y)$, the partial derivative with respect to $x$ tells the rate at which $f$ increases or decreases if only $x$ is allowed to change: $\frac{\partial f}{\partial x}$ In the world beyond math, this might look like asking: How does a plant’s growth respond to extra sunlight if water stays constant? Or, how does the comfort in a room shift if only the temperature rises, while humidity does not budge? The core idea is focus—tuning out every other influence for a moment and seeing the effect of just one change. It’s a small, controlled experiment, repeated for each variable in turn. Partial derivatives offer a way to trace the shape of a complex system, one direction at a time, revealing how each thread is woven into the whole. Notation and Mathematical Definition of Partial Derivatives At first, the notation for partial derivatives might look unfamiliar, but each symbol carries a clear meaning. The most common is the “curly d,” written as $\partial$, which signals that only one variable is changing at a time. For a function $f(x, y)$, the partial derivative with respect to $x$ is written as: $\frac{\partial f}{\partial x}$ This is read as “the partial derivative of $f$ with respect to $x$.” Here, $y$ is held constant—almost as if $y$ has been pressed on pause. The process works the same for $y$: $\frac{\partial f}{\partial y}$ And if there are more variables, the pattern continues. For $f(x, y, z)$, you might see: $\frac{\partial f}{\partial z}$ The Formal Definition Mathematically, the partial derivative is defined as a limit. For $f(x, y)$, the partial derivative with respect to $x$ at a specific point $(a, b)$ is: $\displaystyle \frac{\partial f}{\partial x}(a,b) = \lim_{h \to 0} \frac{f(a+h,,b) - f(a,,b)}{h}$ This formula captures what happens to $f$ as $x$ increases by a tiny amount $h$, while $y$ remains fixed at $b$. No matter how many variables there are, the logic is always the same: only one is allowed to move, the rest are temporarily held steady. With practice, these symbols become a way to trace the changing shape of any landscape, mathematical or otherwise. Higher-Order and Mixed Partial Derivatives Higher-Order Partial Derivatives A higher-order partial derivative is found by taking the partial derivative of a partial derivative. For example, take a function $f(x, y)$. The first partial derivative with respect to $x$ is $\frac{\partial f}{\partial x}$. If this is differentiated again with respect to $x$, the result is the second-order partial derivative: $\frac{\partial^2 f}{\partial x^2}$ This measures how the rate of change with respect to $x$ changes as $x$ continues to change. Imagine climbing a hill: the second-order derivative tells if the slope is getting steeper or more gentle as you move east. Mixed Partial Derivatives Mixed partial derivatives measure how the rate of change with respect to one variable shifts as another variable changes. For $f(x, y)$, the mixed partial derivative with respect to $x$ and then $y$ is: $\frac{\partial^2 f}{\partial y \partial x}$ This means: first, find the partial derivative with respect to $x$, then take the partial derivative of that result with respect to $y$. In real life, this might answer questions like, “How does the effect of more study hours on a test score change as sleep hours increase?” If the function is well-behaved (meaning it is “nice enough,” with no sharp corners or breaks), the mixed partial derivatives can be taken in any order and still get the same answer: $\frac{\partial^2 f}{\partial y \partial x} = \frac{\partial^2 f}{\partial x \partial y}$ Interpreting Partial Derivatives Interpreting a partial derivative means understanding what the value represents in a specific context. For a function $f(x, y)$, the partial derivative $\frac{\partial f}{\partial x}$ describes how much $f$ will increase or decrease if $x$ changes by a tiny amount, while $y$ is held constant. For example, if $f(x, y)$ gives the amount of money earned at a lemonade stand, where $x$ is the number of lemons used and $y$ is the cups of sugar, then $\frac{\partial f}{\partial x}$ tells how much extra money is earned for each extra lemon added—assuming the sugar stays the same. This makes it possible to see, one step at a time, how a single choice affects the result. Real-World Applications of Partial Derivatives Partial derivatives might seem abstract at first glance, but their fingerprints are everywhere in daily life and across nearly every field that measures, predicts, or designs. In economics, $\frac{\partial P}{\partial n}$ measures how profit changes with each additional product sold, keeping advertising constant. In medicine, $\frac{\partial R}{\partial d}$ shows how a patient’s response varies with drug dosage, while sleep remains fixed. In engineering, $\frac{\partial S}{\partial T}$ reveals how material stress shifts with temperature, force held steady. In weather forecasting, $\frac{\partial T}{\partial h}$ tells how temperature changes with altitude at a given location. In environmental science, $\frac{\partial C}{\partial t}$ describes how pollution concentration changes over time at one spot. In machine learning, $\frac{\partial L}{\partial w}$ helps algorithms adjust model weights to improve prediction accuracy. How to Calculate Partial Derivatives Manually Solving partial derivatives by hand is often about careful focus—paying attention to one variable, while treating all others as if they’re constants. Here’s how to approach it, step by step: 1. Pick the Variable Decide which variable you’re differentiating with respect to—call it $x$. All other variables (like $y$, $z$, etc.) are held constant during this step. 2. Differentiate as Usual Apply the rules of differentiation to $x$, just as you would for a single-variable function. If you see a $y$ or $z$, treat it as if it were just a number. 3. Leave Other Variables Alone Variables that aren’t the one you’re differentiating by don’t change. If the term doesn’t contain $x$, its derivative with respect to $x$ is $0$. Example Suppose $f(x, y) = 3x^2y + 2y^2 + 7x$. To find the partial derivative with respect to $x$: $3x^2y$ becomes $6xy$ (since $y$ is treated as constant) $2y^2$ becomes $0$ (no $x$ in this term) $7x$ becomes $7$ So, $\frac{\partial f}{\partial x} = 6xy + 7$ To find the partial derivative with respect to $y$: $3x^2y$ becomes $3x^2$ (since $x$ is treated as constant) $2y^2$ becomes $4y$ $7x$ becomes $0$ So, $\frac{\partial f}{\partial y} = 3x^2 + 4y$ This approach works for any function of two or more variables. Keep your focus on just one variable at a time, and the process becomes much clearer. Calculating partial derivatives manually not only builds understanding, but also strengthens confidence for more complex problems. Common Mistakes and How to Avoid Them Partial derivatives are powerful, but a few classic missteps can trip up even careful students. Here are some common mistakes—plus gentle reminders to help keep your work on track: Forgetting to Hold Other Variables Constant: When differentiating with respect to $x$, every other variable (like $y$ or $z$) should be treated as if it is a fixed number. Only $x$ is “alive” in that moment. Differentiating Terms Without the Chosen Variable: Any term that does not contain the variable you are differentiating with respect to has a derivative of $0$ for that step. For example, in $\frac{\partial}{\partial x}(5y^2)$, the answer is $0$. Mixing Up Notation: Using $\frac{d}{dx}$ instead of $\frac{\partial}{\partial x}$, or forgetting the curly $\partial$, is a small but common slip. Remember, partial derivatives always use the $\partial$ symbol. Dropping Variables Too Early: Don’t remove variables from your answer unless the differentiation process sets them to $0$. For example, $3x^2y$ differentiated with respect to $x$ becomes $6xy$, not just $6x$. Forgetting Which Variable is “In Play”: Always double-check which variable you’re differentiating by, especially in functions with several variables. How to Use the Partial Derivative Calculator on Symbolab When calculations get lengthy or you want extra reassurance, Symbolab’s Partial Derivative Calculator is a supportive tool for every learner. Here’s how to use it, step by step: Step 1: Enter the Expression Type directly using your keyboard: Enter your function, such as $f(x, y) = 3x^2y + 2y^2 + 7x$. Use the math keyboard: Handy for entering square roots, fractions, exponents , and other special math symbols. Upload a photo: Snap a picture of a handwritten problem or textbook page. Use the Chrome extension: Capture math directly from a webpage by highlighting and screenshotting the expression. Press the “Go” button. Step 2: View Step-by-Step Breakdown Watch as each part of the solution is explained clearly, with the option to go through one step at a time. If a step doesn’t make sense, or if you’re curious about a detail, use the ‘Chat with Symbo’ to ask questions or get clarifications tailored to your needs. Conclusion Partial derivatives are more than a set of rules—they are a way of seeing how each piece of a bigger puzzle contributes to change. Whether solved by hand or explored with Symbolab’s calculator, each calculation reveals a little more about how the world’s variables interact. With patience and practice, these tools help make complex situations understandable, guiding thoughtful decisions in science, business, engineering, or everyday life. Frequently Asked Questions (FAQ) How do you find the partial derivative? To calculate the partial derivative of a function choose the variable with respect to which you want to take the partial derivative, and treat all the other variables as constant. Differentiate the function with respect to the chosen variable, using the rules of differentiation. What are the rules of partial derivatives? Partial derivatives follow the sane rules as derivatives: the sum rule, the difference rule, the product rule, the quotient rule, and the chain rule. What is the sum rule of partial derivatives? The sum rule of partial derivatives is a technique for calculating the partial derivative of the sum of two functions. It states that if f(x,y) and g(x,y) are both differentiable functions, then: ∂(f+g)/∂x = ∂f/∂x + ∂g/∂x ∂(f+g)/∂y = ∂f/∂y + ∂g/∂y What is the difference rule of partial derivatives? The difference rule of partial derivatives is a technique for calculating the partial derivative of the difference of two functions. It states that if f(x,y) and g(x,y) are both differentiable functions, then: ∂(f-g)/∂x = ∂f/∂x - ∂g/∂x ∂(f-g)/∂y = ∂f/∂y - ∂g/∂y What is the product rule of partial derivatives? The product rule of partial derivatives is a technique for calculating the partial derivative of the product of two functions. It states that if f(x,y) and g(x,y) are both differentiable functions, then: ∂(fg)/∂x = f∂g/∂x + g∂f/∂x ∂(fg)/∂y = f∂g/∂y + g∂f/∂y What is the quotient rule of partial derivatives? The quotient rule of partial derivatives is a technique for calculating the partial derivative of the quotient of two functions. It states that if f(x,y) and g(x,y) are both differentiable functions and g(x,y) is not equal to 0, then: ∂(f/g)/∂x = (∂f/∂xg - f∂g/∂x)/g^2 ∂(f/g)/∂y = (∂f/∂yg - f∂g/∂y)/g^2 What is the chain rule of partial derivatives? The chain rule of partial derivatives is a technique for calculating the partial derivative of a composite function. It states that if f(x,y) and g(x,y) are both differentiable functions, and y is a function of x (i.e. y = h(x)), then: ∂f/∂x = ∂f/∂y * ∂y/∂x What is the partial derivative of a function? The partial derivative of a function is a way of measuring how much the function changes when you change one of its variables, while holding the other variables constant.

Markdown

You can see your coupon in the [user page](https://www.symbolab.com/user#subscription) [Go To QuillBot Upgrade to Pro Continue to site]() [We've updated our]() [Privacy Policy](https://www.symbolab.com/privacy-policy) effective December 15. Please read our updated Privacy Policy and tap Continue - [Solutions](https://www.symbolab.com/solver) [Integral Calculator](https://www.symbolab.com/solver/integral-calculator) [Derivative Calculator](https://www.symbolab.com/solver/derivative-calculator) [Algebra Calculator](https://www.symbolab.com/solver/algebra-calculator) [Matrix Calculator](https://www.symbolab.com/solver/matrix-calculator) [More...](https://www.symbolab.com/solver) - [Graphing](https://www.symbolab.com/graphing-calculator) [Line Graph Calculator](https://www.symbolab.com/graphing-calculator/line) [Exponential Graph Calculator](https://www.symbolab.com/graphing-calculator/exponential-graph) [Quadratic Graph Calculator](https://www.symbolab.com/graphing-calculator/quadratic-graph) [Sine Graph Calculator](https://www.symbolab.com/graphing-calculator/sine-graph) [More...](https://www.symbolab.com/graphing-calculator#examples=true) - [Calculators](https://www.symbolab.com/calculators) [BMI Calculator](https://www.symbolab.com/calculator/fitness/bmi) [Compound Interest Calculator](https://www.symbolab.com/calculator/finance/compound_interest_rate) [Percentage Calculator](https://www.symbolab.com/calculator/math/percentage) [Acceleration Calculator](https://www.symbolab.com/calculator/physics/acceleration) [More...](https://www.symbolab.com/calculators) - [Geometry](https://www.symbolab.com/geometry-calculator) [Pythagorean Theorem Calculator](https://www.symbolab.com/geometry-calculator/pythagorean-theorem-calculator) [Circle Area Calculator](https://www.symbolab.com/geometry-calculator/circle-area-diameter-calculator) [Isosceles Triangle Calculator](https://www.symbolab.com/geometry-calculator/isosceles-triangle-prove-angles-calculator) [Triangles Calculator](https://www.symbolab.com/geometry-calculator/triangle-perimeter-calculator) [More...](https://www.symbolab.com/geometry-calculator#examples=true) - [AI Chat](https://www.symbolab.com/ai-chat) - [Tools](https://www.symbolab.com/solver/partial-derivative-calculator) [Notebook](https://www.symbolab.com/notebook) [Groups](https://www.symbolab.com/groups) [Cheat Sheets](https://www.symbolab.com/cheat-sheets) [Worksheets](https://www.symbolab.com/worksheets/Pre-Algebra) [Study Guides](https://www.symbolab.com/study-guides) [Practice](https://www.symbolab.com/practice) [Verify Solution](https://www.symbolab.com/verify) - [Upgrade](https://www.symbolab.com/solver/partial-derivative-calculator) Close Accept Terms Good job\! ![Practice](https://www.symbolab.com/public/images/practice_icon1.png) Practice More Type your Answer **Verify** | | | | | | | | | | | | | |---|---|---|---|---|---|---|---|---|---|---|---| | | | | | | | | | | | | | | x^2 | x^{\\msquare} | \\log\_{\\msquare} | \\sqrt{\\square} | \\nthroot\[\\msquare\]{\\square} | \\le | \\ge | \\frac{\\msquare}{\\msquare} | \\cdot | \\div | x^{\\circ} | \\pi | | \\left(\\square\\right)^{'} | \\frac{d}{dx} | \\frac{\\partial}{\\partial x} | \\int | \\int\_{\\msquare}^{\\msquare} | \\lim | \\sum | \\infty | \\theta | (f\\:\\circ\\:g) | f(x) | ![](https://www.symbolab.com/public/img/pad_matrixnxn.png) | | | | | | | | | | |---|---|---|---|---|---|---|---| | ▭\\:\\longdivision{▭} | \\times \\twostack{▭}{▭} | \+ \\twostack{▭}{▭} | \- \\twostack{▭}{▭} | \\left( | \\right) | \\times | \\square\\frac{\\square}{\\square} | Take a challenge Subscribe to verify your answer Subscribe Are you sure you want to leave this Challenge? By closing this window you will lose this challenge Cancel Leave - [Pre Algebra](https://www.symbolab.com/solver/pre-algebra-calculator) [Order of Operations](https://www.symbolab.com/solver/order-of-operations-calculator) [Factors & Primes](https://www.symbolab.com/solver/factors-primes-calculator) [Fractions](https://www.symbolab.com/solver/fractions-calculator) [Long Arithmetic](https://www.symbolab.com/solver/long-arithmetic-calculator) [Decimals](https://www.symbolab.com/solver/decimals-calculator) [Exponents & Radicals](https://www.symbolab.com/solver/exponents-radicals-calculator) [Ratios & Proportions](https://www.symbolab.com/solver/ratios-proportions-calculator) [Percent](https://www.symbolab.com/solver/percent-calculator) [Modulo](https://www.symbolab.com/solver/modulo-calculator) [Number Line](https://www.symbolab.com/solver/number-line-calculator) [Expanded Form](https://www.symbolab.com/solver/expanded-form-calculator) [Mean, Median & Mode](https://www.symbolab.com/solver/mean-median-mode-calculator) - [Algebra](https://www.symbolab.com/solver/algebra-calculator) [Equations](https://www.symbolab.com/solver/equation-calculator) [Inequalities](https://www.symbolab.com/solver/inequalities-calculator) [System of Equations](https://www.symbolab.com/solver/system-of-equations-calculator) [System of Inequalities](https://www.symbolab.com/solver/system-of-inequalities-calculator) [Testing Solutions](https://www.symbolab.com/solver/testing-solutions-calculator) [Basic Operations](https://www.symbolab.com/solver/basic-operations-calculator) [Algebraic Properties](https://www.symbolab.com/solver/algebraic-properties-calculator) [Partial Fractions](https://www.symbolab.com/solver/partial-fractions-calculator) [Polynomials](https://www.symbolab.com/solver/polynomial-calculator) [Rational Expressions](https://www.symbolab.com/solver/rational-expression-calculator) [Sequences](https://www.symbolab.com/solver/sequence-calculator) [Power Sums](https://www.symbolab.com/solver/power-sums-calculator) [Interval Notation](https://www.symbolab.com/solver/interval-notation-calculator) [Pi (Product) Notation](https://www.symbolab.com/solver/pi-product-notation-calculator) [Induction](https://www.symbolab.com/solver/induction-calculator) [Prove That](https://www.symbolab.com/solver/prove-that-calculator) [Logical Sets](https://www.symbolab.com/solver/logical-sets-calculator) [Word Problems](https://www.symbolab.com/solver/word-problems-calculator) - [Pre Calculus](https://www.symbolab.com/solver/pre-calculus-calculator) [Equations](https://www.symbolab.com/solver/pre-calculus-equation-calculator) [Inequalities](https://www.symbolab.com/solver/pre-calculus-inequalities-calculator) [Scientific Calculator](https://www.symbolab.com/solver/scientific-calculator) [Scientific Notation Arithmetics](https://www.symbolab.com/solver/scientific-notation-arithmetics) [Complex Numbers](https://www.symbolab.com/solver/pre-calculus-complex-numbers-calculator) [Polar/Cartesian](https://www.symbolab.com/solver/polar-cartesian-calculator) [Simultaneous Equations](https://www.symbolab.com/solver/simultaneous-equations-calculator) [System of Inequalities](https://www.symbolab.com/solver/pre-calculus-system-of-inequalities-calculator) [Polynomials](https://www.symbolab.com/solver/pre-calculus-polynomial-calculator) [Rationales](https://www.symbolab.com/solver/rationales-calculator) [Functions](https://www.symbolab.com/solver/pre-calculus-functions-calculator) [Arithmetic & Comp.](https://www.symbolab.com/solver/pre-calculus-function-arithmetic-composition-calculator) [Coordinate Geometry](https://www.symbolab.com/solver/pre-calculus-coordinate-geometry-calculator) [Plane Geometry](https://www.symbolab.com/solver/plane-geometry-calculator) [Solid Geometry](https://www.symbolab.com/solver/solid-geometry-calculator) [Trigonometry](https://www.symbolab.com/solver/pre-calculus-trigonometry-calculator) - [Calculus](https://www.symbolab.com/solver/calculus-calculator) [Derivatives](https://www.symbolab.com/solver/derivative-calculator) [Derivative Applications](https://www.symbolab.com/solver/derivative-applications-calculator) [Limits](https://www.symbolab.com/solver/limit-calculator) [Integrals](https://www.symbolab.com/solver/integral-calculator) [Integral Applications](https://www.symbolab.com/solver/integral-applications-calculator) [Integral Approximation](https://www.symbolab.com/solver/integral-approximation-calculator) [Series](https://www.symbolab.com/solver/series-calculator) [ODE](https://www.symbolab.com/solver/ordinary-differential-equation-calculator) [Multivariable Calculus](https://www.symbolab.com/solver/multivariable-calculus-calculator) [Laplace Transform](https://www.symbolab.com/solver/laplace-calculator) [Taylor/Maclaurin Series](https://www.symbolab.com/solver/taylor-maclaurin-series-calculator) [Fourier Series](https://www.symbolab.com/solver/fourier-series-calculator) [Fourier Transform](https://www.symbolab.com/solver/fourier-transform-calculator) - [Functions](https://www.symbolab.com/solver/functions-line-calculator) [Line Equations](https://www.symbolab.com/solver/line-equation-calculator) [Functions](https://www.symbolab.com/solver/functions-calculator) [Arithmetic & Comp.](https://www.symbolab.com/solver/function-arithmetic-composition-calculator) [Conic Sections](https://www.symbolab.com/solver/conic-sections-calculator) [Transformation](https://www.symbolab.com/solver/function-transformation-calculator) - [Linear Algebra](https://www.symbolab.com/solver/linear-algebra-calculator) [Matrices](https://www.symbolab.com/solver/matrix-calculator) [Vectors](https://www.symbolab.com/solver/vector-calculator) - [Trigonometry](https://www.symbolab.com/solver/trigonometry-calculator) [Quadrant](https://www.symbolab.com/solver/quadrant-calculator) [Coterminal Angle](https://www.symbolab.com/solver/coterminal-angle-calculator) [Identities](https://www.symbolab.com/solver/trigonometric-identity-calculator) [Proving Identities](https://www.symbolab.com/solver/trigonometric-identity-proving-calculator) [Trig Equations](https://www.symbolab.com/solver/trigonometric-equation-calculator) [Trig Inequalities](https://www.symbolab.com/solver/trigonometric-inequality-calculator) [Evaluate Functions](https://www.symbolab.com/solver/trigonometric-function-calculator) [Simplify](https://www.symbolab.com/solver/trigonometric-simplification-calculator) - [Statistics](https://www.symbolab.com/solver/statistics-calculator) [Mean](https://www.symbolab.com/solver/arithmetic-mean-calculator) [Geometric Mean](https://www.symbolab.com/solver/geometric-mean-calculator) [Quadratic Mean](https://www.symbolab.com/solver/quadratic-mean-calculator) [Average](https://www.symbolab.com/solver/average-calculator) [Median](https://www.symbolab.com/solver/median-calculator) [Mode](https://www.symbolab.com/solver/mode-calculator) [Order](https://www.symbolab.com/solver/order-calculator) [Minimum](https://www.symbolab.com/solver/minimum-calculator) [Maximum](https://www.symbolab.com/solver/maximum-calculator) [Probability](https://www.symbolab.com/solver/probability-calculator) [Mid-Range](https://www.symbolab.com/solver/mid-range-calculator) [Range](https://www.symbolab.com/solver/range-calculator) [Standard Deviation](https://www.symbolab.com/solver/standard-deviation-calculator) [Variance](https://www.symbolab.com/solver/variance-calculator) [Lower Quartile](https://www.symbolab.com/solver/lower-quartile-calculator) [Upper Quartile](https://www.symbolab.com/solver/upper-quartile-calculator) [Interquartile Range](https://www.symbolab.com/solver/interquartile-range-calculator) [Midhinge](https://www.symbolab.com/solver/midhinge-calculator) [Standard Normal Distribution](https://www.symbolab.com/solver/standard-normal-distribution-calculator) - [Physics](https://www.symbolab.com/solver/physics-calculator) [Mechanics](https://www.symbolab.com/solver/mechanics-calculator) - [Chemistry](https://www.symbolab.com/solver/chemistry-calculator) [Chemical Reactions](https://www.symbolab.com/solver/chemical-reaction-calculator) [Chemical Properties](https://www.symbolab.com/solver/chemical-properties-calculator) - [Finance](https://www.symbolab.com/solver/finance-calculator) [Simple Interest](https://www.symbolab.com/solver/simple-interest-calculator) [Compound Interest](https://www.symbolab.com/solver/compound-interest-calculator) [Present Value](https://www.symbolab.com/solver/present-value-calculator) [Future Value](https://www.symbolab.com/solver/future-value-calculator) - [Economics](https://www.symbolab.com/solver/economics-calculator) [Point of Diminishing Return](https://www.symbolab.com/solver/point-of-diminishing-return-calculator) - [Conversions](https://www.symbolab.com/solver/conversion-calculator) [Currency](https://www.symbolab.com/solver/currency-converter) [Roman Numerals](https://www.symbolab.com/solver/roman-numeral-converter) [Radical to Exponent](https://www.symbolab.com/solver/radical-to-exponent-calculator) [Exponent to Radical](https://www.symbolab.com/solver/exponent-to-radical-calculator) [To Fraction](https://www.symbolab.com/solver/convert-to-fraction-calculator) [To Decimal](https://www.symbolab.com/solver/convert-to-decimal-calculator) [To Mixed Number](https://www.symbolab.com/solver/convert-to-mixed-number-calculator) [To Improper Fraction](https://www.symbolab.com/solver/convert-to-improper-fraction-calculator) [Radians to Degrees](https://www.symbolab.com/solver/radians-to-degrees-calculator) [Degrees to Radians](https://www.symbolab.com/solver/degrees-to-radians-calculator) [Degrees Minutes Seconds](https://www.symbolab.com/solver/degrees-minutes-seconds-calculator) [Hexadecimal](https://www.symbolab.com/solver/hexadecimal-converter) [Scientific Notation](https://www.symbolab.com/solver/scientific-notation-converter) [Distance](https://www.symbolab.com/solver/distance-converter) [Weight](https://www.symbolab.com/solver/weight-converter) [Time](https://www.symbolab.com/solver/time-converter) [Volume](https://www.symbolab.com/solver/volume-converter) ### Special Offer ## Double the Tools. One Smart Bundle. to master your math and writing\! Pro - Full access to solution steps - Full access to AI chat - Practice and improve - Access from any device - No ads - Unlimited storage - Detect AI-generated content - Advanced Grammar Checker - AI Humanizer - Paraphrase in unlimited modes - Summarize any text - Much More\! [Restrictions apply, check out our FAQs.](https://www.symbolab.com/bundle-faq) Go back Purchase Bundle ### Back to School Promotion Annual Annual - \$ % OFF Annual plan One time offer for one year, then \$ Go back \+ ## qb-banner-title ### ## ### [Solutions](https://www.symbolab.com/solver) \> [Calculus Calculator](https://www.symbolab.com/solver/calculus-calculator) \> # Partial Derivative Calculator Topic - [Pre Algebra](https://www.symbolab.com/solver/pre-algebra-calculator) - [Algebra](https://www.symbolab.com/solver/algebra-calculator) - [Pre Calculus](https://www.symbolab.com/solver/pre-calculus-calculator) - [Calculus](https://www.symbolab.com/solver/calculus-calculator) - [Derivatives]() - [First Derivative](https://www.symbolab.com/solver/first-derivative-calculator) - [WRT](https://www.symbolab.com/solver/derivative-with-respect-calculator) - [Specify Method]() - [Chain Rule](https://www.symbolab.com/solver/derivative-chain-rule-calculator) - [Product Rule](https://www.symbolab.com/solver/derivative-product-rule-calculator) - [Quotient Rule](https://www.symbolab.com/solver/derivative-quotient-rule-calculator) - [Sum/Diff Rule](https://www.symbolab.com/solver/derivative-sum-diff-rule-calculator) - [Second Derivative](https://www.symbolab.com/solver/second-derivative-calculator) - [Third Derivative](https://www.symbolab.com/solver/third-derivative-calculator) - [Higher Order Derivatives](https://www.symbolab.com/solver/higher-order-derivative-calculator) - [Derivative at a point](https://www.symbolab.com/solver/derivative-point-calculator) - [Partial Derivative](https://www.symbolab.com/solver/partial-derivative-calculator) - [Implicit Derivative](https://www.symbolab.com/solver/implicit-derivative-calculator) - [Second Implicit Derivative](https://www.symbolab.com/solver/second-implicit-derivative-calculator) - [Derivative using Definition](https://www.symbolab.com/solver/derivative-using-definition-calculator) - [Derivative Applications]() - [Tangent](https://www.symbolab.com/solver/tangent-line-calculator) - [Slope of Tangent](https://www.symbolab.com/solver/slope-of-tangent-calculator) - [Normal](https://www.symbolab.com/solver/normal-line-calculator) - [Curved Line Slope](https://www.symbolab.com/solver/curved-line-slope-calculator) - [Extreme Points](https://www.symbolab.com/solver/calculus-function-extreme-points-calculator) - [Tangent to Conic](https://www.symbolab.com/solver/tangent-to-conic-calculator) - [Linear Approximation](https://www.symbolab.com/solver/linear-approximation-calculator) - [Difference Quotient](https://www.symbolab.com/solver/difference-quotient-calculator) - [Horizontal Tangent](https://www.symbolab.com/solver/horizontal-tangent-calculator) - [Limits]() - [One Variable](https://www.symbolab.com/solver/single-var-limit-calculator) - [Multi Variable Limit](https://www.symbolab.com/solver/multi-var-limit-calculator) - [One Sided](https://www.symbolab.com/solver/one-sided-limit-calculator) - [At Infinity](https://www.symbolab.com/solver/limit-infinity-calculator) - [Specify Method]() - [L'Hopital's Rule](https://www.symbolab.com/solver/limit-lhopital-calculator) - [Squeeze Theorem](https://www.symbolab.com/solver/limit-squeeze-theorem-calculator) - [Chain Rule](https://www.symbolab.com/solver/limit-chain-rule-calculator) - [Factoring](https://www.symbolab.com/solver/limit-using-factoring-calculator) - [Substitution](https://www.symbolab.com/solver/limit-substitution-calculator) - [Sandwich Theorem](https://www.symbolab.com/solver/limit-sandwich-theorem-calculator) - [Integrals]() - [Indefinite Integrals](https://www.symbolab.com/solver/indefinite-integral-calculator) - [Definite Integrals](https://www.symbolab.com/solver/definite-integral-calculator) - [Specific-Method]() - [Partial Fractions](https://www.symbolab.com/solver/partial-fractions-integration-calculator) - [U-Substitution](https://www.symbolab.com/solver/u-substitution-integration-calculator) - [Trigonometric Substitution](https://www.symbolab.com/solver/trigonometric-substitution-integration-calculator) - [Weierstrass Substitution](https://www.symbolab.com/solver/weierstrass-substitution-calculator) - [By Parts](https://www.symbolab.com/solver/by-parts-integration-calculator) - [Long Division](https://www.symbolab.com/solver/long-division-integration-calculator) - [Improper Integrals](https://www.symbolab.com/solver/improper-integral-calculator) - [Antiderivatives](https://www.symbolab.com/solver/antiderivative-calculator) - [Double Integrals](https://www.symbolab.com/solver/double-integrals-calculator) - [Triple Integrals](https://www.symbolab.com/solver/triple-integrals-calculator) - [Multiple Integrals](https://www.symbolab.com/solver/multiple-integrals-calculator) - [Integral Applications]() - [Limit of Sum](https://www.symbolab.com/solver/limit-of-sum-calculator) - [Area under curve](https://www.symbolab.com/solver/area-under-curve-calculator) - [Area between curves](https://www.symbolab.com/solver/area-between-curves-calculator) - [Area under polar curve](https://www.symbolab.com/solver/area-under-polar-curve-calculator) - [Volume of solid of revolution](https://www.symbolab.com/solver/volume-calculator) - [Arc Length](https://www.symbolab.com/solver/arc-length-calculator) - [Function Average](https://www.symbolab.com/solver/function-average-calculator) - [Integral Approximation]() - [Riemann Sum](https://www.symbolab.com/solver/riemann-sum-calculator) - [Trapezoidal](https://www.symbolab.com/solver/trapezoidal-approximation-calculator) - [Simpson's Rule](https://www.symbolab.com/solver/simpsons-rule-calculator) - [Midpoint Rule](https://www.symbolab.com/solver/midpoint-rule-calculator) - [Series]() - [Convergence]() - [Geometric Series Test](https://www.symbolab.com/solver/geometric-series-test-calculator) - [Telescoping Series Test](https://www.symbolab.com/solver/telescoping-series-test-calculator) - [Alternating Series Test](https://www.symbolab.com/solver/alternating-series-test-calculator) - [P Series Test](https://www.symbolab.com/solver/p-series-test-calculator) - [Divergence Test](https://www.symbolab.com/solver/series-divergence-test-calculator) - [Ratio Test](https://www.symbolab.com/solver/series-ratio-test-calculator) - [Root Test](https://www.symbolab.com/solver/series-root-test-calculator) - [Comparison Test](https://www.symbolab.com/solver/series-comparison-test-calculator) - [Limit Comparison Test](https://www.symbolab.com/solver/series-limit-comparison-test-calculator) - [Integral Test](https://www.symbolab.com/solver/series-integral-test-calculator) - [Absolute Convergence](https://www.symbolab.com/solver/series-absolute-convergence-calculator) - [Power Series]() - [Radius of Convergence](https://www.symbolab.com/solver/radius-of-convergence-calculator) - [Interval of Convergence](https://www.symbolab.com/solver/interval-of-convergence-calculator) - [ODE]() - [Linear First Order](https://www.symbolab.com/solver/linear-first-order-differential-equation-calculator) - [Linear w/constant coefficients](https://www.symbolab.com/solver/linear-constant-coefficients-differential-equation-calculator) - [Separable](https://www.symbolab.com/solver/separable-differential-equation-calculator) - [Bernoulli](https://www.symbolab.com/solver/bernoulli-differential-equation-calculator) - [Exact](https://www.symbolab.com/solver/exact-differential-equation-calculator) - [Second Order](https://www.symbolab.com/solver/second-order-differential-equation-calculator) - [Homogenous](https://www.symbolab.com/solver/homogenous-differential-equation-calculator) - [Non Homogenous](https://www.symbolab.com/solver/non-homogenous-differential-equation-calculator) - [Substitution](https://www.symbolab.com/solver/substitution-differential-equation-calculator) - [System of ODEs](https://www.symbolab.com/solver/system-of-differential-equations-calculator) - [IVP using Laplace](https://www.symbolab.com/solver/ivp-laplace-calculator) - [Series Solutions](https://www.symbolab.com/solver/ode-series-solutions-calculator) - [Method of Frobenius](https://www.symbolab.com/solver/frobenius-method-calculator) - [Gamma Function](https://www.symbolab.com/solver/gamma-function-calculator) - [Multivariable Calculus]() - [Partial Derivative](https://www.symbolab.com/solver/multivariable-partial-derivative-calculator) - [Implicit Derivative](https://www.symbolab.com/solver/multivariable-implicit-derivative-calculator) - [Tangent to Conic](https://www.symbolab.com/solver/multivariable-tangent-to-conic-calculator) - [Multi Variable Limit](https://www.symbolab.com/solver/multi-var-calculus-limit-calculator) - [Multiple Integrals](https://www.symbolab.com/solver/multiple-integrals-multivariable-calculator) - [Gradient](https://www.symbolab.com/solver/gradient-calculator) - [Divergence](https://www.symbolab.com/solver/divergence-calculator) - [Extreme Points](https://www.symbolab.com/solver/multi-var-function-extreme-points-calculator) - [Laplace Transform]() - [Inverse](https://www.symbolab.com/solver/inverse-laplace-calculator) - [Taylor/Maclaurin Series]() - [Taylor Series](https://www.symbolab.com/solver/taylor-series-calculator) - [Maclaurin Series](https://www.symbolab.com/solver/maclaurin-series-calculator) - [Fourier Series](https://www.symbolab.com/solver/fourier-series-calculator) - [Fourier Transform](https://www.symbolab.com/solver/fourier-transform-calculator) - [Functions](https://www.symbolab.com/solver/functions-line-calculator) - [Linear Algebra](https://www.symbolab.com/solver/linear-algebra-calculator) - [Trigonometry](https://www.symbolab.com/solver/trigonometry-calculator) - [Statistics](https://www.symbolab.com/solver/statistics-calculator) - [Physics](https://www.symbolab.com/solver/physics-calculator) - [Chemistry](https://www.symbolab.com/solver/chemistry-calculator) - [Finance](https://www.symbolab.com/solver/finance-calculator) - [Economics](https://www.symbolab.com/solver/economics-calculator) - [Conversions](https://www.symbolab.com/solver/conversion-calculator) Add to Chrome Get our extension, you can capture any math problem from any website Full pad | | | | | | | | | | | | | |---|---|---|---|---|---|---|---|---|---|---|---| | | | | | | | | | | | | | | x^2 | x^{\\msquare} | \\log\_{\\msquare} | \\sqrt{\\square} | \\nthroot\[\\msquare\]{\\square} | \\le | \\ge | \\frac{\\msquare}{\\msquare} | \\cdot | \\div | x^{\\circ} | \\pi | | \\left(\\square\\right)^{'} | \\frac{d}{dx} | \\frac{\\partial}{\\partial x} | \\int | \\int\_{\\msquare}^{\\msquare} | \\lim | \\sum | \\infty | \\theta | (f\\:\\circ\\:g) | f(x) | ![](https://www.symbolab.com/public/img/pad_matrixnxn.png) | | | | | | | | | |---|---|---|---|---|---|---| | \- \\twostack{▭}{▭} | \\lt | 7 | 8 | 9 | \\div | AC | | \+ \\twostack{▭}{▭} | \\gt | 4 | 5 | 6 | \\times | \\square\\frac{\\square}{\\square} | | \\times \\twostack{▭}{▭} | \\left( | 1 | 2 | 3 | \- | x | | ▭\\:\\longdivision{▭} | \\right) | . | 0 | \= | \+ | y | \\mathrm{implicit\\:derivative} \\mathrm{tangent} \\mathrm{volume} \\mathrm{laplace} \\mathrm{fourier} See All area asymptotes critical points derivative domain eigenvalues eigenvectors expand extreme points factor implicit derivative inflection points intercepts inverse laplace inverse laplace partial fractions range slope simplify solve for tangent taylor vertex geometric test alternating test telescoping test pseries test root test **Go** [Steps](https://www.symbolab.com/solver/partial-derivative-calculator#steps) [Graph](https://www.symbolab.com/solver/partial-derivative-calculator#plot) [Related](https://www.symbolab.com/solver/partial-derivative-calculator#related_anchor) [Examples](https://www.symbolab.com/solver/partial-derivative-calculator#Examples) Generated by AI AI explanations are generated using OpenAI technology. AI generated content may present inaccurate or offensive content that does not represent Symbolab's view. Verify your Answer Subscribe to verify your answer Subscribe Save to Notebook\! Sign in to save notes Sign in Verify Save Show Steps Hide Steps ## Number Line ![]() ![](https://www.symbolab.com/public/img/ajax-loader.gif) Related Partial Derivative Examples - [\\frac{\\partial}{\\partial x}(\\sin (x^2y^2))](https://www.symbolab.com/solver/partial-derivative-calculator/%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20x%7D%28%5Csin%20%28x%5E2y%5E2%29%29?or=ex) - [\\frac{\\partial}{\\partial y}(\\sin (x^2y^2))](https://www.symbolab.com/solver/partial-derivative-calculator/%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20y%7D%28%5Csin%20%28x%5E2y%5E2%29%29?or=ex) - [\\frac{\\partial}{\\partial y\\partial x}(\\sin (x^2y^2))](https://www.symbolab.com/solver/partial-derivative-calculator/%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20y%5Cpartial%20x%7D%28%5Csin%20%28x%5E2y%5E2%29%29?or=ex) - [\\frac{\\partial}{\\partial w}(te^{(\\frac{w}{t})})](https://www.symbolab.com/solver/partial-derivative-calculator/%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20w%7D%28te%5E%7B%28%5Cfrac%7Bw%7D%7Bt%7D%29%7D%29?or=ex) - [\\frac{\\partial}{\\partial t}(te^{(\\frac{w}{t})})](https://www.symbolab.com/solver/partial-derivative-calculator/%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20t%7D%28te%5E%7B%28%5Cfrac%7Bw%7D%7Bt%7D%29%7D%29?or=ex) - [\\frac{\\partial}{\\partial v}(\\sqrt{u^2+v^2})](https://www.symbolab.com/solver/partial-derivative-calculator/%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20v%7D%28%5Csqrt%7Bu%5E2%2Bv%5E2%7D%29?or=ex) - Show More ## A Step-by-Step Guide to Partial Derivatives Calculator Many moments in daily life hinge on more than one changing factor. The taste of homemade soup, for example, is not just about the mix of spices but also the time the pot spends simmering. A perfect photograph is shaped by both the [angle](https://www.symbolab.com/geometry-calculator/angles-calculator) of the light and the split-second timing of the shutter. There’s rarely only one ingredient at play. When several [variables](https://www.symbolab.com/solver/solve-for-equation-calculator) come together to shape an outcome, partial derivatives offer a quiet way to notice: what happens if only one piece of the puzzle moves, while everything else stands still? In this article, partial derivatives will be explored one careful step at a time—what they are, why they matter, how they show up in daily life, and how to work with them using Symbolab’s Partial Derivative Calculator. ## What are Partial Derivatives Think of any outcome that depends on a handful of choices, not just one. In math, these relationships are described by [functions](https://www.symbolab.com/solver/functions-calculator) like \$f(x, y)\$, where both \$x\$ and \$y\$ are inputs—sometimes these are hours studied and hours slept, sometimes they’re temperature and humidity, sometimes they’re the amount of flour and the oven’s heat. Each factor quietly pushes and pulls the result in its own way. A partial derivative asks: if one variable changes—say, \$x\$—while \$y\$ stays exactly the same, how does the outcome respond? It’s like gently turning one dial on a stereo and listening for just that note in the music. Mathematically, the partial derivative with respect to \$x\$ is written as: \$\\frac{\\partial f}{\\partial x}\$ and the same question about \$y\$ gives: \$\\frac{\\partial f}{\\partial y}\$ No need to let the symbols overwhelm; the heart of it is this: only one input shifts, the rest are frozen in place. For something even more down-to-earth, think about cooking rice. The final texture, \$T\$, depends on both water \$w\$ and heat \$h\$—so \$T(w, h)\$. Suppose you want to know how the rice’s texture changes when you increase the heat while keeping the water level constant. That rate of change is captured by the partial derivative with respect to heat, written as: \$\\frac{\\partial T}{\\partial h}\$ measures just that. How does the rice change, with water unchanged, as the flame turns up? This idea isn’t only for kitchens or science labs; it appears wherever complex outcomes depend on several choices at once. Partial derivatives let us slow down and focus, revealing how small, single changes can shape a bigger story. They’re a first, essential step toward making sense of the whole beautiful tangle of variables that fill real life. ## Understanding Multivariable Functions It’s rare for one single variable to hold all the power. Most real-world outcomes—the brightness of a room, the total at a checkout counter, the time it takes to bike across town—depend on two or more factors working together. In math, multivariable functions capture these relationships. A multivariable function might look like \$f(x, y)\$, where both \$x\$ and \$y\$ influence the result. Think about a weather app that predicts temperature based on time of day (\$t\$) and cloud cover (\$c\$), or a bakery’s profit, \$P(c, p)\$, that depends on the number of cakes (\$c\$) and pies (\$p\$) sold. Each variable tugs on the outcome in its own way. For example: In a classroom, a student’s test score might depend on hours spent studying (\$s\$) and hours slept the night before (\$h\$): \$S(s, h)\$. A family’s grocery bill could be modeled as \$B(f, v)\$, where \$f\$ is pounds of fruit and \$v\$ is pounds of vegetables in the cart. Sometimes, these functions are simple sums: \$f(x,y)=2x+3y\$ Other times, they weave together in more interesting patterns, like \$f(x, y) = x^2 + xy + y^2\$ or \$f(x, y, z) = xyz\$ Multivariable functions create a kind of landscape—each combination of inputs gives a new point in that landscape. Understanding how each variable shapes the outcome, one at a time, is where partial derivatives enter the scene. But before calculating changes, it helps to pause and notice: in math as in life, the most interesting results often depend on more than one ingredient. ## The Concept of Partial Derivatives Picture a hill stretching out in every direction. The height at any point depends on how far north or east you walk. This is a classic multivariable function—height as a function of \$x\$ (east-west) and \$y\$ (north-south): \$h(x, y)\$. Now, imagine standing still in the north-south direction, and taking a small step east. The question becomes: how does the height change as you move only east, with your north-south position unchanged? That is the spirit of a partial derivative. It measures how a function’s output shifts as you change one input ever so slightly, while all other variables are frozen. In mathematical language, for a function \$f(x, y)\$, the partial derivative with respect to \$x\$ tells the rate at which \$f\$ increases or decreases if only \$x\$ is allowed to change: \$\\frac{\\partial f}{\\partial x}\$ In the world beyond math, this might look like asking: How does a plant’s growth respond to extra sunlight if water stays constant? Or, how does the comfort in a room shift if only the temperature rises, while humidity does not budge? The core idea is focus—tuning out every other influence for a moment and seeing the effect of just one change. It’s a small, controlled experiment, repeated for each variable in turn. Partial derivatives offer a way to trace the shape of a complex system, one direction at a time, revealing how each thread is woven into the whole. ## Notation and Mathematical Definition of Partial Derivatives At first, the notation for partial derivatives might look unfamiliar, but each symbol carries a clear meaning. The most common is the “curly d,” written as \$\\partial\$, which signals that only one variable is changing at a time. For a function \$f(x, y)\$, the partial derivative with respect to \$x\$ is written as: \$\\frac{\\partial f}{\\partial x}\$ This is read as “the partial derivative of \$f\$ with respect to \$x\$.” Here, \$y\$ is held constant—almost as if \$y\$ has been pressed on pause. The process works the same for \$y\$: \$\\frac{\\partial f}{\\partial y}\$ And if there are more variables, the pattern continues. For \$f(x, y, z)\$, you might see: \$\\frac{\\partial f}{\\partial z}\$ ### The Formal Definition Mathematically, the partial derivative is defined as a limit. For \$f(x, y)\$, the partial derivative with respect to \$x\$ at a specific point \$(a, b)\$ is: \$\\displaystyle \\frac{\\partial f}{\\partial x}(a,b) = \\lim\_{h \\to 0} \\frac{f(a+h,,b) - f(a,,b)}{h}\$ This formula captures what happens to \$f\$ as \$x\$ increases by a tiny amount \$h\$, while \$y\$ remains fixed at \$b\$. No matter how many variables there are, the logic is always the same: only one is allowed to move, the rest are temporarily held steady. With practice, these symbols become a way to trace the changing shape of any landscape, mathematical or otherwise. ## Higher-Order and Mixed Partial Derivatives ### Higher-Order Partial Derivatives A higher-order partial derivative is found by taking the partial derivative of a partial derivative. For example, take a function \$f(x, y)\$. The first partial derivative with respect to \$x\$ is \$\\frac{\\partial f}{\\partial x}\$. If this is differentiated again with respect to \$x\$, the result is the second-order partial derivative: \$\\frac{\\partial^2 f}{\\partial x^2}\$ This measures how the rate of change with respect to \$x\$ changes as \$x\$ continues to change. Imagine climbing a hill: the second-order derivative tells if the slope is getting steeper or more gentle as you move east. ### Mixed Partial Derivatives Mixed partial derivatives measure how the rate of change with respect to one variable shifts as another variable changes. For \$f(x, y)\$, the mixed partial derivative with respect to \$x\$ and then \$y\$ is: \$\\frac{\\partial^2 f}{\\partial y \\partial x}\$ This means: first, find the partial derivative with respect to \$x\$, then take the partial derivative of that result with respect to \$y\$. In real life, this might answer questions like, “How does the effect of more study hours on a test score change as sleep hours increase?” If the function is well-behaved (meaning it is “nice enough,” with no sharp corners or breaks), the mixed partial derivatives can be taken in any order and still get the same answer: \$\\frac{\\partial^2 f}{\\partial y \\partial x} = \\frac{\\partial^2 f}{\\partial x \\partial y}\$ ## Interpreting Partial Derivatives Interpreting a partial derivative means understanding what the value represents in a specific context. For a function \$f(x, y)\$, the partial derivative \$\\frac{\\partial f}{\\partial x}\$ describes how much \$f\$ will increase or decrease if \$x\$ changes by a tiny amount, while \$y\$ is held constant. For example, if \$f(x, y)\$ gives the amount of money earned at a lemonade stand, where \$x\$ is the number of lemons used and \$y\$ is the cups of sugar, then \$\\frac{\\partial f}{\\partial x}\$ tells how much extra money is earned for each extra lemon added—assuming the sugar stays the same. This makes it possible to see, one step at a time, how a single choice affects the result. ## Real-World Applications of Partial Derivatives Partial derivatives might seem abstract at first glance, but their fingerprints are everywhere in daily life and across nearly every field that measures, predicts, or designs. - In economics, \$\\frac{\\partial P}{\\partial n}\$ measures how profit changes with each additional product sold, keeping advertising constant. - In medicine, \$\\frac{\\partial R}{\\partial d}\$ shows how a patient’s response varies with drug dosage, while sleep remains fixed. - In engineering, \$\\frac{\\partial S}{\\partial T}\$ reveals how material stress shifts with temperature, force held steady. - In weather forecasting, \$\\frac{\\partial T}{\\partial h}\$ tells how temperature changes with altitude at a given location. - In environmental science, \$\\frac{\\partial C}{\\partial t}\$ describes how pollution concentration changes over time at one spot. - In machine learning, \$\\frac{\\partial L}{\\partial w}\$ helps algorithms adjust model weights to improve prediction accuracy. ## How to Calculate Partial Derivatives Manually Solving partial derivatives by hand is often about careful focus—paying attention to one variable, while treating all others as if they’re constants. Here’s how to approach it, step by step: ### 1\. Pick the Variable Decide which variable you’re differentiating with respect to—call it \$x\$. All other variables (like \$y\$, \$z\$, etc.) are held constant during this step. ### 2\. Differentiate as Usual Apply the rules of differentiation to \$x\$, just as you would for a single-variable function. If you see a \$y\$ or \$z\$, treat it as if it were just a number. ### 3\. Leave Other Variables Alone Variables that aren’t the one you’re differentiating by don’t change. If the term doesn’t contain \$x\$, its [derivative](https://www.symbolab.com/solver/derivative-calculator) with respect to \$x\$ is \$0\$. Example Suppose \$f(x, y) = 3x^2y + 2y^2 + 7x\$. To find the partial derivative with respect to \$x\$: - \$3x^2y\$ becomes \$6xy\$ (since \$y\$ is treated as constant) - \$2y^2\$ becomes \$0\$ (no \$x\$ in this term) - \$7x\$ becomes \$7\$ So, \$\\frac{\\partial f}{\\partial x} = 6xy + 7\$ To find the partial derivative with respect to \$y\$: - \$3x^2y\$ becomes \$3x^2\$ (since \$x\$ is treated as constant) - \$2y^2\$ becomes \$4y\$ - \$7x\$ becomes \$0\$ So, \$\\frac{\\partial f}{\\partial y} = 3x^2 + 4y\$ This approach works for any function of two or more variables. Keep your focus on just one variable at a time, and the process becomes much clearer. Calculating partial derivatives manually not only builds understanding, but also strengthens confidence for more complex problems. ## Common Mistakes and How to Avoid Them Partial derivatives are powerful, but a few classic missteps can trip up even careful students. Here are some common mistakes—plus gentle reminders to help keep your work on track: - **Forgetting to Hold Other Variables Constant:** When differentiating with respect to \$x\$, every other variable (like \$y\$ or \$z\$) should be treated as if it is a fixed number. Only \$x\$ is “alive” in that moment. - **Differentiating Terms Without the Chosen Variable:** Any term that does not contain the variable you are differentiating with respect to has a derivative of \$0\$ for that step. For example, in \$\\frac{\\partial}{\\partial x}(5y^2)\$, the answer is \$0\$. - **Mixing Up Notation:** Using \$\\frac{d}{dx}\$ instead of \$\\frac{\\partial}{\\partial x}\$, or forgetting the curly \$\\partial\$, is a small but common slip. Remember, partial derivatives always use the \$\\partial\$ symbol. - **Dropping Variables Too Early:** Don’t remove variables from your answer unless the differentiation process sets them to \$0\$. For example, \$3x^2y\$ differentiated with respect to \$x\$ becomes \$6xy\$, not just \$6x\$. - **Forgetting Which Variable is “In Play”:** Always double-check which variable you’re differentiating by, especially in functions with several variables. ## How to Use the Partial Derivative Calculator on Symbolab When calculations get lengthy or you want extra reassurance, Symbolab’s Partial Derivative Calculator is a supportive tool for every learner. Here’s how to use it, step by step: ### Step 1: Enter the Expression - Type directly using your keyboard: Enter your function, such as \$f(x, y) = 3x^2y + 2y^2 + 7x\$. - Use the math keyboard: Handy for entering square roots, fractions, [exponents](https://www.symbolab.com/solver/exponents-calculator), and other special math symbols. - Upload a photo: Snap a picture of a handwritten problem or textbook page. - Use the Chrome extension: Capture math directly from a webpage by highlighting and screenshotting the expression. Press the “Go” button. ### Step 2: View Step-by-Step Breakdown Watch as each part of the solution is explained clearly, with the option to go through one step at a time. If a step doesn’t make sense, or if you’re curious about a detail, use the ‘Chat with Symbo’ to ask questions or get clarifications tailored to your needs. ## Conclusion Partial derivatives are more than a set of rules—they are a way of seeing how each piece of a bigger puzzle contributes to change. Whether solved by hand or explored with Symbolab’s calculator, each calculation reveals a little more about how the world’s variables interact. With patience and practice, these tools help make complex situations understandable, guiding thoughtful decisions in science, business, engineering, or everyday life. Frequently Asked Questions (FAQ) - How do you find the partial derivative? - To calculate the partial derivative of a function choose the variable with respect to which you want to take the partial derivative, and treat all the other variables as constant. Differentiate the function with respect to the chosen variable, using the rules of differentiation. - What are the rules of partial derivatives? - Partial derivatives follow the sane rules as derivatives: the sum rule, the difference rule, the product rule, the quotient rule, and the chain rule. - What is the sum rule of partial derivatives? - The sum rule of partial derivatives is a technique for calculating the partial derivative of the sum of two functions. It states that if f(x,y) and g(x,y) are both differentiable functions, then: ∂(f+g)/∂x = ∂f/∂x + ∂g/∂x ∂(f+g)/∂y = ∂f/∂y + ∂g/∂y - What is the difference rule of partial derivatives? - The difference rule of partial derivatives is a technique for calculating the partial derivative of the difference of two functions. It states that if f(x,y) and g(x,y) are both differentiable functions, then: ∂(f-g)/∂x = ∂f/∂x - ∂g/∂x ∂(f-g)/∂y = ∂f/∂y - ∂g/∂y - What is the product rule of partial derivatives? - The product rule of partial derivatives is a technique for calculating the partial derivative of the product of two functions. It states that if f(x,y) and g(x,y) are both differentiable functions, then: ∂(fg)/∂x = f∂g/∂x + g∂f/∂x ∂(fg)/∂y = f∂g/∂y + g∂f/∂y - What is the quotient rule of partial derivatives? - The quotient rule of partial derivatives is a technique for calculating the partial derivative of the quotient of two functions. It states that if f(x,y) and g(x,y) are both differentiable functions and g(x,y) is not equal to 0, then: ∂(f/g)/∂x = (∂f/∂xg - f∂g/∂x)/g^2 ∂(f/g)/∂y = (∂f/∂yg - f∂g/∂y)/g^2 - What is the chain rule of partial derivatives? - The chain rule of partial derivatives is a technique for calculating the partial derivative of a composite function. It states that if f(x,y) and g(x,y) are both differentiable functions, and y is a function of x (i.e. y = h(x)), then: ∂f/∂x = ∂f/∂y \* ∂y/∂x - What is the partial derivative of a function? - The partial derivative of a function is a way of measuring how much the function changes when you change one of its variables, while holding the other variables constant. partial-derivative-calculator en Related Symbolab blog posts [High School Math Solutions – Derivative Calculator, the Basics Differentiation is a method to calculate the rate of change (or the slope at a point on the graph); we will not...](https://blog.symbolab.com/2014/01/high-school-math-solutions-derivative.html) Popular topics [scientific calculator](https://www.symbolab.com/solver/scientific-calculator) [inverse calculator](https://www.symbolab.com/solver/function-inverse-calculator) [simplify calculator](https://www.symbolab.com/solver/simplify-calculator) [distance calculator](https://www.symbolab.com/solver/distance-calculator) [fractions calculator](https://www.symbolab.com/solver/fractions-calculator) [interval notation calculator](https://www.symbolab.com/solver/interval-notation-calculator) [cross product calculator](https://www.symbolab.com/solver/vector-cross-product-calculator) [probability calculator](https://www.symbolab.com/solver/probability-calculator) [derivative calculator](https://www.symbolab.com/solver/derivative-calculator) [series calculator](https://www.symbolab.com/solver/series-calculator) [ratios calculator](https://www.symbolab.com/solver/ratios-calculator) [statistics calculator](https://www.symbolab.com/solver/statistics-calculator) [integral calculator](https://www.symbolab.com/solver/integral-calculator) [inverse laplace transform calculator](https://www.symbolab.com/solver/inverse-laplace-calculator) [rounding calculator](https://www.symbolab.com/solver/rounding-calculator) [gcf calculator](https://www.symbolab.com/solver/gcf-calculator) [algebra calculator](https://www.symbolab.com/solver/algebra-calculator) [tangent line calculator](https://www.symbolab.com/solver/tangent-line-calculator) [trigonometry calculator](https://www.symbolab.com/solver/trigonometry-calculator) [log calculator](https://www.symbolab.com/solver/logarithmic-equation-calculator) [standard deviation calculator](https://www.symbolab.com/solver/standard-deviation-calculator) [linear equation calculator](https://www.symbolab.com/solver/linear-equation-calculator) [antiderivative calculator](https://www.symbolab.com/solver/antiderivative-calculator) [laplace transform calculator](https://www.symbolab.com/solver/laplace-calculator) [quadratic equation calculator](https://www.symbolab.com/solver/quadratic-equation-calculator) [domain calculator](https://www.symbolab.com/solver/function-domain-calculator) [decimals calculator](https://www.symbolab.com/solver/decimals-calculator) [limit calculator](https://www.symbolab.com/solver/limit-calculator) [equation solver](https://www.symbolab.com/solver/equation-calculator) [definite integral calculator](https://www.symbolab.com/solver/definite-integral-calculator) [matrix inverse calculator](https://www.symbolab.com/solver/matrix-inverse-calculator) [matrix calculator](https://www.symbolab.com/solver/matrix-calculator) [system of equations calculator](https://www.symbolab.com/solver/system-of-equations-calculator) [calculus calculator](https://www.symbolab.com/solver/calculus-calculator) [slope calculator](https://www.symbolab.com/solver/slope-calculator) [long division calculator](https://www.symbolab.com/solver/polynomial-long-division-calculator) [factors calculator](https://www.symbolab.com/solver/factor-calculator) [polynomial calculator](https://www.symbolab.com/solver/polynomial-equation-calculator) [square root calculator](https://www.symbolab.com/solver/square-root-calculator) [implicit differentiation calculator](https://www.symbolab.com/solver/implicit-derivative-calculator) [word problem solver](https://www.symbolab.com/solver/word-problems-calculator) [differential equation calculator](https://www.symbolab.com/solver/ordinary-differential-equation-calculator) [average calculator](https://www.symbolab.com/calculator/statistics/average) [synthetic division calculator](https://www.symbolab.com/solver/synthetic-division-calculator) Chat with Symbo AI AI may present inaccurate or offensive content that does not represent Symbolab's views. Do not enter any personal information Enter a problem [Cooking Calculators](https://www.symbolab.com/calculator/cooking) [Cooking Measurement Converter](https://www.symbolab.com/calculator/cooking/cooking-measurement) [Cooking Ingredient Converter](https://www.symbolab.com/calculator/cooking/cooking-ingredient) [Cake Pan Converter](https://www.symbolab.com/calculator/cooking/cake-pan) [More calculators](https://www.symbolab.com/calculator/cooking) [Fitness Calculators](https://www.symbolab.com/calculator/fitness) [BMI Calculator](https://www.symbolab.com/calculator/fitness/bmi) [Calorie Calculator](https://www.symbolab.com/calculator/fitness/calorie) [BMR Calculator](https://www.symbolab.com/calculator/fitness/bmr) [More calculators](https://www.symbolab.com/calculator/fitness) Save to Notebook\! Sign in Notebook [View Full Notebook](https://www.symbolab.com/notebook) Study Tools [AI Math Solver](https://www.symbolab.com/) [AI Chat](https://www.symbolab.com/ai-chat) [Popular Problems](https://www.symbolab.com/popular-algebra) [Worksheets](https://www.symbolab.com/worksheets/Pre-Algebra) [Study Guides](https://www.symbolab.com/study-guides) [Practice](https://www.symbolab.com/practice) [Cheat Sheets](https://www.symbolab.com/cheat-sheets) [Calculators](https://www.symbolab.com/calculators) [Graphing Calculator](https://www.symbolab.com/graphing-calculator) [Geometry Calculator](https://www.symbolab.com/geometry-calculator) [Verify Solution](https://www.symbolab.com/verify) Apps [Symbolab App (Android)](https://play.google.com/store/apps/details?id=com.devsense.symbolab) [Graphing Calculator (Android)](https://play.google.com/store/apps/details?id=com.symbolab.graphingcalculator) [Practice (Android)](https://play.google.com/store/apps/details?id=com.symbolab.practice) [Symbolab App (iOS)](https://apps.apple.com/us/app/id876942533) [Graphing Calculator (iOS)](https://apps.apple.com/us/app/id1435182798) [Practice (iOS)](https://apps.apple.com/us/app/id1469186281) [Chrome Extension](https://chromewebstore.google.com/detail/mgfcnnpakbfcefgphceolkjldjfpieff) Company [About Symbolab](https://www.symbolab.com/about) [Blog](https://blog.symbolab.com/) [Help](https://www.symbolab.com/help) [Contact Us](https://www.symbolab.com/solver/partial-derivative-calculator) Legal [Privacy](https://www.symbolab.com/privacy-policy) [Service Terms](https://www.symbolab.com/terms-of-use) [Cookie Policy](https://www.symbolab.com/cookie-policy) Cookie Settings Copyright, Community Guidelines, DSA & other Legal Resources [Learneo Legal Center](https://learneo.com/legal) [Feedback](https://www.symbolab.com/solver/partial-derivative-calculator) Social Media Symbolab, a Learneo, Inc. business © Learneo, Inc. 2024 *** (optional) (optional) Please add a message. Message received. Thanks for the feedback. Cancel **Send** ![](https://www.symbolab.com/public/img/ajax-loader.gif) ![]()

Readable Markdown

## A Step-by-Step Guide to Partial Derivatives Calculator Many moments in daily life hinge on more than one changing factor. The taste of homemade soup, for example, is not just about the mix of spices but also the time the pot spends simmering. A perfect photograph is shaped by both the [angle](https://www.symbolab.com/geometry-calculator/angles-calculator) of the light and the split-second timing of the shutter. There’s rarely only one ingredient at play. When several [variables](https://www.symbolab.com/solver/solve-for-equation-calculator) come together to shape an outcome, partial derivatives offer a quiet way to notice: what happens if only one piece of the puzzle moves, while everything else stands still? In this article, partial derivatives will be explored one careful step at a time—what they are, why they matter, how they show up in daily life, and how to work with them using Symbolab’s Partial Derivative Calculator. ## What are Partial Derivatives Think of any outcome that depends on a handful of choices, not just one. In math, these relationships are described by [functions](https://www.symbolab.com/solver/functions-calculator) like \$f(x, y)\$, where both \$x\$ and \$y\$ are inputs—sometimes these are hours studied and hours slept, sometimes they’re temperature and humidity, sometimes they’re the amount of flour and the oven’s heat. Each factor quietly pushes and pulls the result in its own way. A partial derivative asks: if one variable changes—say, \$x\$—while \$y\$ stays exactly the same, how does the outcome respond? It’s like gently turning one dial on a stereo and listening for just that note in the music. Mathematically, the partial derivative with respect to \$x\$ is written as: \$\\frac{\\partial f}{\\partial x}\$ and the same question about \$y\$ gives: \$\\frac{\\partial f}{\\partial y}\$ No need to let the symbols overwhelm; the heart of it is this: only one input shifts, the rest are frozen in place. For something even more down-to-earth, think about cooking rice. The final texture, \$T\$, depends on both water \$w\$ and heat \$h\$—so \$T(w, h)\$. Suppose you want to know how the rice’s texture changes when you increase the heat while keeping the water level constant. That rate of change is captured by the partial derivative with respect to heat, written as: \$\\frac{\\partial T}{\\partial h}\$ measures just that. How does the rice change, with water unchanged, as the flame turns up? This idea isn’t only for kitchens or science labs; it appears wherever complex outcomes depend on several choices at once. Partial derivatives let us slow down and focus, revealing how small, single changes can shape a bigger story. They’re a first, essential step toward making sense of the whole beautiful tangle of variables that fill real life. ## Understanding Multivariable Functions It’s rare for one single variable to hold all the power. Most real-world outcomes—the brightness of a room, the total at a checkout counter, the time it takes to bike across town—depend on two or more factors working together. In math, multivariable functions capture these relationships. A multivariable function might look like \$f(x, y)\$, where both \$x\$ and \$y\$ influence the result. Think about a weather app that predicts temperature based on time of day (\$t\$) and cloud cover (\$c\$), or a bakery’s profit, \$P(c, p)\$, that depends on the number of cakes (\$c\$) and pies (\$p\$) sold. Each variable tugs on the outcome in its own way. For example: In a classroom, a student’s test score might depend on hours spent studying (\$s\$) and hours slept the night before (\$h\$): \$S(s, h)\$. A family’s grocery bill could be modeled as \$B(f, v)\$, where \$f\$ is pounds of fruit and \$v\$ is pounds of vegetables in the cart. Sometimes, these functions are simple sums: \$f(x,y)=2x+3y\$ Other times, they weave together in more interesting patterns, like \$f(x, y) = x^2 + xy + y^2\$ or \$f(x, y, z) = xyz\$ Multivariable functions create a kind of landscape—each combination of inputs gives a new point in that landscape. Understanding how each variable shapes the outcome, one at a time, is where partial derivatives enter the scene. But before calculating changes, it helps to pause and notice: in math as in life, the most interesting results often depend on more than one ingredient. ## The Concept of Partial Derivatives Picture a hill stretching out in every direction. The height at any point depends on how far north or east you walk. This is a classic multivariable function—height as a function of \$x\$ (east-west) and \$y\$ (north-south): \$h(x, y)\$. Now, imagine standing still in the north-south direction, and taking a small step east. The question becomes: how does the height change as you move only east, with your north-south position unchanged? That is the spirit of a partial derivative. It measures how a function’s output shifts as you change one input ever so slightly, while all other variables are frozen. In mathematical language, for a function \$f(x, y)\$, the partial derivative with respect to \$x\$ tells the rate at which \$f\$ increases or decreases if only \$x\$ is allowed to change: \$\\frac{\\partial f}{\\partial x}\$ In the world beyond math, this might look like asking: How does a plant’s growth respond to extra sunlight if water stays constant? Or, how does the comfort in a room shift if only the temperature rises, while humidity does not budge? The core idea is focus—tuning out every other influence for a moment and seeing the effect of just one change. It’s a small, controlled experiment, repeated for each variable in turn. Partial derivatives offer a way to trace the shape of a complex system, one direction at a time, revealing how each thread is woven into the whole. ## Notation and Mathematical Definition of Partial Derivatives At first, the notation for partial derivatives might look unfamiliar, but each symbol carries a clear meaning. The most common is the “curly d,” written as \$\\partial\$, which signals that only one variable is changing at a time. For a function \$f(x, y)\$, the partial derivative with respect to \$x\$ is written as: \$\\frac{\\partial f}{\\partial x}\$ This is read as “the partial derivative of \$f\$ with respect to \$x\$.” Here, \$y\$ is held constant—almost as if \$y\$ has been pressed on pause. The process works the same for \$y\$: \$\\frac{\\partial f}{\\partial y}\$ And if there are more variables, the pattern continues. For \$f(x, y, z)\$, you might see: \$\\frac{\\partial f}{\\partial z}\$ ### The Formal Definition Mathematically, the partial derivative is defined as a limit. For \$f(x, y)\$, the partial derivative with respect to \$x\$ at a specific point \$(a, b)\$ is: \$\\displaystyle \\frac{\\partial f}{\\partial x}(a,b) = \\lim\_{h \\to 0} \\frac{f(a+h,,b) - f(a,,b)}{h}\$ This formula captures what happens to \$f\$ as \$x\$ increases by a tiny amount \$h\$, while \$y\$ remains fixed at \$b\$. No matter how many variables there are, the logic is always the same: only one is allowed to move, the rest are temporarily held steady. With practice, these symbols become a way to trace the changing shape of any landscape, mathematical or otherwise. ## Higher-Order and Mixed Partial Derivatives ### Higher-Order Partial Derivatives A higher-order partial derivative is found by taking the partial derivative of a partial derivative. For example, take a function \$f(x, y)\$. The first partial derivative with respect to \$x\$ is \$\\frac{\\partial f}{\\partial x}\$. If this is differentiated again with respect to \$x\$, the result is the second-order partial derivative: \$\\frac{\\partial^2 f}{\\partial x^2}\$ This measures how the rate of change with respect to \$x\$ changes as \$x\$ continues to change. Imagine climbing a hill: the second-order derivative tells if the slope is getting steeper or more gentle as you move east. ### Mixed Partial Derivatives Mixed partial derivatives measure how the rate of change with respect to one variable shifts as another variable changes. For \$f(x, y)\$, the mixed partial derivative with respect to \$x\$ and then \$y\$ is: \$\\frac{\\partial^2 f}{\\partial y \\partial x}\$ This means: first, find the partial derivative with respect to \$x\$, then take the partial derivative of that result with respect to \$y\$. In real life, this might answer questions like, “How does the effect of more study hours on a test score change as sleep hours increase?” If the function is well-behaved (meaning it is “nice enough,” with no sharp corners or breaks), the mixed partial derivatives can be taken in any order and still get the same answer: \$\\frac{\\partial^2 f}{\\partial y \\partial x} = \\frac{\\partial^2 f}{\\partial x \\partial y}\$ ## Interpreting Partial Derivatives Interpreting a partial derivative means understanding what the value represents in a specific context. For a function \$f(x, y)\$, the partial derivative \$\\frac{\\partial f}{\\partial x}\$ describes how much \$f\$ will increase or decrease if \$x\$ changes by a tiny amount, while \$y\$ is held constant. For example, if \$f(x, y)\$ gives the amount of money earned at a lemonade stand, where \$x\$ is the number of lemons used and \$y\$ is the cups of sugar, then \$\\frac{\\partial f}{\\partial x}\$ tells how much extra money is earned for each extra lemon added—assuming the sugar stays the same. This makes it possible to see, one step at a time, how a single choice affects the result. ## Real-World Applications of Partial Derivatives Partial derivatives might seem abstract at first glance, but their fingerprints are everywhere in daily life and across nearly every field that measures, predicts, or designs. - In economics, \$\\frac{\\partial P}{\\partial n}\$ measures how profit changes with each additional product sold, keeping advertising constant. - In medicine, \$\\frac{\\partial R}{\\partial d}\$ shows how a patient’s response varies with drug dosage, while sleep remains fixed. - In engineering, \$\\frac{\\partial S}{\\partial T}\$ reveals how material stress shifts with temperature, force held steady. - In weather forecasting, \$\\frac{\\partial T}{\\partial h}\$ tells how temperature changes with altitude at a given location. - In environmental science, \$\\frac{\\partial C}{\\partial t}\$ describes how pollution concentration changes over time at one spot. - In machine learning, \$\\frac{\\partial L}{\\partial w}\$ helps algorithms adjust model weights to improve prediction accuracy. ## How to Calculate Partial Derivatives Manually Solving partial derivatives by hand is often about careful focus—paying attention to one variable, while treating all others as if they’re constants. Here’s how to approach it, step by step: ### 1\. Pick the Variable Decide which variable you’re differentiating with respect to—call it \$x\$. All other variables (like \$y\$, \$z\$, etc.) are held constant during this step. ### 2\. Differentiate as Usual Apply the rules of differentiation to \$x\$, just as you would for a single-variable function. If you see a \$y\$ or \$z\$, treat it as if it were just a number. ### 3\. Leave Other Variables Alone Variables that aren’t the one you’re differentiating by don’t change. If the term doesn’t contain \$x\$, its [derivative](https://www.symbolab.com/solver/derivative-calculator) with respect to \$x\$ is \$0\$. Example Suppose \$f(x, y) = 3x^2y + 2y^2 + 7x\$. To find the partial derivative with respect to \$x\$: - \$3x^2y\$ becomes \$6xy\$ (since \$y\$ is treated as constant) - \$2y^2\$ becomes \$0\$ (no \$x\$ in this term) - \$7x\$ becomes \$7\$ So, \$\\frac{\\partial f}{\\partial x} = 6xy + 7\$ To find the partial derivative with respect to \$y\$: - \$3x^2y\$ becomes \$3x^2\$ (since \$x\$ is treated as constant) - \$2y^2\$ becomes \$4y\$ - \$7x\$ becomes \$0\$ So, \$\\frac{\\partial f}{\\partial y} = 3x^2 + 4y\$ This approach works for any function of two or more variables. Keep your focus on just one variable at a time, and the process becomes much clearer. Calculating partial derivatives manually not only builds understanding, but also strengthens confidence for more complex problems. ## Common Mistakes and How to Avoid Them Partial derivatives are powerful, but a few classic missteps can trip up even careful students. Here are some common mistakes—plus gentle reminders to help keep your work on track: - **Forgetting to Hold Other Variables Constant:** When differentiating with respect to \$x\$, every other variable (like \$y\$ or \$z\$) should be treated as if it is a fixed number. Only \$x\$ is “alive” in that moment. - **Differentiating Terms Without the Chosen Variable:** Any term that does not contain the variable you are differentiating with respect to has a derivative of \$0\$ for that step. For example, in \$\\frac{\\partial}{\\partial x}(5y^2)\$, the answer is \$0\$. - **Mixing Up Notation:** Using \$\\frac{d}{dx}\$ instead of \$\\frac{\\partial}{\\partial x}\$, or forgetting the curly \$\\partial\$, is a small but common slip. Remember, partial derivatives always use the \$\\partial\$ symbol. - **Dropping Variables Too Early:** Don’t remove variables from your answer unless the differentiation process sets them to \$0\$. For example, \$3x^2y\$ differentiated with respect to \$x\$ becomes \$6xy\$, not just \$6x\$. - **Forgetting Which Variable is “In Play”:** Always double-check which variable you’re differentiating by, especially in functions with several variables. ## How to Use the Partial Derivative Calculator on Symbolab When calculations get lengthy or you want extra reassurance, Symbolab’s Partial Derivative Calculator is a supportive tool for every learner. Here’s how to use it, step by step: ### Step 1: Enter the Expression - Type directly using your keyboard: Enter your function, such as \$f(x, y) = 3x^2y + 2y^2 + 7x\$. - Use the math keyboard: Handy for entering square roots, fractions, [exponents](https://www.symbolab.com/solver/exponents-calculator), and other special math symbols. - Upload a photo: Snap a picture of a handwritten problem or textbook page. - Use the Chrome extension: Capture math directly from a webpage by highlighting and screenshotting the expression. Press the “Go” button. ### Step 2: View Step-by-Step Breakdown Watch as each part of the solution is explained clearly, with the option to go through one step at a time. If a step doesn’t make sense, or if you’re curious about a detail, use the ‘Chat with Symbo’ to ask questions or get clarifications tailored to your needs. ## Conclusion Partial derivatives are more than a set of rules—they are a way of seeing how each piece of a bigger puzzle contributes to change. Whether solved by hand or explored with Symbolab’s calculator, each calculation reveals a little more about how the world’s variables interact. With patience and practice, these tools help make complex situations understandable, guiding thoughtful decisions in science, business, engineering, or everyday life. Frequently Asked Questions (FAQ) - How do you find the partial derivative? - To calculate the partial derivative of a function choose the variable with respect to which you want to take the partial derivative, and treat all the other variables as constant. Differentiate the function with respect to the chosen variable, using the rules of differentiation. - What are the rules of partial derivatives? - Partial derivatives follow the sane rules as derivatives: the sum rule, the difference rule, the product rule, the quotient rule, and the chain rule. - What is the sum rule of partial derivatives? - The sum rule of partial derivatives is a technique for calculating the partial derivative of the sum of two functions. It states that if f(x,y) and g(x,y) are both differentiable functions, then: ∂(f+g)/∂x = ∂f/∂x + ∂g/∂x ∂(f+g)/∂y = ∂f/∂y + ∂g/∂y - What is the difference rule of partial derivatives? - The difference rule of partial derivatives is a technique for calculating the partial derivative of the difference of two functions. It states that if f(x,y) and g(x,y) are both differentiable functions, then: ∂(f-g)/∂x = ∂f/∂x - ∂g/∂x ∂(f-g)/∂y = ∂f/∂y - ∂g/∂y - What is the product rule of partial derivatives? - The product rule of partial derivatives is a technique for calculating the partial derivative of the product of two functions. It states that if f(x,y) and g(x,y) are both differentiable functions, then: ∂(fg)/∂x = f∂g/∂x + g∂f/∂x ∂(fg)/∂y = f∂g/∂y + g∂f/∂y - What is the quotient rule of partial derivatives? - The quotient rule of partial derivatives is a technique for calculating the partial derivative of the quotient of two functions. It states that if f(x,y) and g(x,y) are both differentiable functions and g(x,y) is not equal to 0, then: ∂(f/g)/∂x = (∂f/∂xg - f∂g/∂x)/g^2 ∂(f/g)/∂y = (∂f/∂yg - f∂g/∂y)/g^2 - What is the chain rule of partial derivatives? - The chain rule of partial derivatives is a technique for calculating the partial derivative of a composite function. It states that if f(x,y) and g(x,y) are both differentiable functions, and y is a function of x (i.e. y = h(x)), then: ∂f/∂x = ∂f/∂y \* ∂y/∂x - What is the partial derivative of a function? - The partial derivative of a function is a way of measuring how much the function changes when you change one of its variables, while holding the other variables constant.

ML Classification

ML Categories

/Science		82.0%
/Science/Mathematics		81.5%
/Science/Mathematics/Other		69.0%
/Computers_and_Electronics		24.9%
/Computers_and_Electronics/Software		18.2%
/Computers_and_Electronics/Software/Educational_Software		13.9%

Raw JSON

{
    "/Science": 820,
    "/Science/Mathematics": 815,
    "/Science/Mathematics/Other": 690,
    "/Computers_and_Electronics": 249,
    "/Computers_and_Electronics/Software": 182,
    "/Computers_and_Electronics/Software/Educational_Software": 139
}

ML Page Types

/Article		61.0%
/Article/Tutorial_or_Guide		57.6%

Raw JSON

{
    "/Article": 610,
    "/Article/Tutorial_or_Guide": 576
}

ML Intent Types

Informational		98.6%
Commercial		11.6%

Raw JSON

{
    "Informational": 986,
    "Commercial": 116
}

Content Metadata

Language

Author

null

Publish Time

not set

Original Publish Time

2015-01-19 19:03:30 (11 years ago)

Republished

Word Count (Total)

3,933

Word Count (Content)

2,669

Links

External Links

Internal Links

324

Technical SEO

Meta Nofollow

Meta Noarchive

JS Rendered

Redirect Target

null

Performance

Download Time (ms)

105

TTFB (ms)

Download Size (bytes)

177,754

Shard

2 (laksa)

Root Hash

1977249391401099202

Unparsed URL

com,symbolab!www,/solver/partial-derivative-calculator s443