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Calculated Shard: 185 (from laksa119)

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curl -X POST \
  'http://laksa185.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 FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://www.cuemath.com/calculus/partial-derivative/\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.cuemath.com/calculus/partial-derivative/\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/www.cuemath.com\/calculus\/partial-derivative\/","crawl_time":1776086523,"first_indexed_time":1651203167,"http_code":200,"src_unparsed":"com,cuemath!www,\/calculus\/partial-derivative\/ s443","src_root_hash":"6203651625873809585","history_drop_reason":null,"meta_title":"Partial Derivative (Partial Differentiation) - Calculate, Symbol","meta_descriptions":["The partial derivative of a function with respect to x is the derivative the function treating all other variables than x as constants. Learn how to calculate the partial derivative along with examples and formulas."],"attrs_boilerpipe_text":"The partial derivative of a function (in two or more variables) is its derivative with respect to one of the variables keeping all the other variables as constants. The process of calculating partial derivative is as same as that of an ordinary derivative except we consider the other variables than the variable with respect to which we are differentiating as constants.\nLet us learn more about how to calculate partial derivatives of different orders along with examples.\n1.\nWhat is Partial Derivative?\n2.\nPartial Derivative Symbol\n3.\nHow to Calculate Partial Derivatives?\n4.\nPartial Derivatives of Different Orders\n5.\nPartial Differentiation Formulas\n6.\nFAQs on Partial Derivative\nThe\npartial derivative\nof a multivariable function, say z = f(x, y), is its\nderivative\nwith respect to one of the variables, x or y in this case, where the other variables are treated as constants. For example,\nfor finding the partial derivative of f(x, y) with respect to x (which is represented by ∂f \/ ∂x), y is treated as constant and\nfor finding the partial derivative of f(x, y) with respect to y (which is represented by ∂f \/ ∂y), x is treated as constant\nNote that we are not considering all the variables as variables while doing partial differentiation (instead, we are considering only one variable as a variable at a time) and hence the name \"partial\". The limit definition of a partial derivative looks very similar to the\nlimit\ndefinition of the derivative. We can find the partial derivatives using the following limit formulas:\n∂f \/ ∂x = lim\nh → 0\n[ f(x + h, y) - f(x, y) ] \/ h\n∂f \/ ∂y = lim\nh → 0\n[ f(x, y + h) - f(x, y) ] \/ h\nThese formulas resemble the derivative definition using the first principle.\nExample of Partial Derivative\nIf f (x, y) = xy, then find the partial derivative ∂f \/ ∂x.\nSolution:\n∂f \/ ∂x = lim\nh → 0\n[ f(x + h, y) - f(x, y) ] \/ h\n= lim\nh → 0\n[ (x + h)y - xy ] \/ h\n= lim\nh → 0\n[xy + hy - xy] \/ h\n= lim\nh → 0\n[hy]\/h\n= lim\nh → 0\ny\n= y\nTherefore, ∂f \/ ∂x = y.\nWe know that the ordinary derivative of a function y = f(x) is denoted by one of the notations dy\/dx, d\/dx (y), d\/dx (f(x)), f '(x), etc. For representing a partial derivative we use the symbol \"∂\" instead of \"d\". We pronounce \"∂\" to be \"doh\" but it has some other names like \"partial\", \"del\", \"partial dee\", \"dee\", \"Jacobi's delta\", etc. If z = f(x, y) is a function in two variables then\n∂f \/ ∂x is the partial derivative of f with respect to x\n∂f \/ ∂y is the partial derivative of f with respect to y\nJust like how we have different\nsymbols\nof ordinary derivatives, we have different notations for partial derivatives as well. For example, ∂f \/ ∂x can be written as f\nx\n, f\nx\n', D\nx\nf, ∂ \/ ∂x (f), ∂\nx\nf, ∂ \/ ∂x [f(x, y)], ∂z \/ ∂x, etc.\nWe have already seen that the limit definitions are used to find the partial derivatives. But using the\nlimit formula\nand computing the limit is not always easy. Thus, we have another method to calculate partial derivatives that follow right from its definition. In this method, if z = f(x, y) is the function, then we can compute the partial derivatives using the following steps:\nStep 1\n: Identify the variable with respect to which we have to find the partial derivative.\nStep 2:\nExcept for the variable found in\nStep 1\n, treat all the other variables as constants.\nStep 3:\nDifferentiate the function just using the rules of ordinary differentiation.\nWait! Read\nStep 3\nagain. Yes, the rules of ordinary differentiation are as same as that of partial differentiation. In partial differentiation, just treating variables is different, that's it!\nExample:\nLet us solve the same above example (If f (x, y) = xy, then find the partial derivative ∂f \/ ∂x) using the above steps.\nSolution:\nWe have to find ∂f \/ ∂x. It means, we have to find the partial derivative of f with respect to x. So we treat y as constant. Thus, we can write 'y' outside the derivative (as in ordinary differentiation, we have a rule that says d\/dx (c y) = c dy\/dx, where 'c' is a constant). Thus,\n∂f \/ ∂x = ∂ \/ ∂x (xy)\n= y ∂ \/ ∂x (x)\n= y (1) (Using\npower rule\n, d\/dx (x) = 1)\n= y\nWe have got the same answer as we got using the limit definition.\nWe have derivatives like first-order derivatives (like dy\/dx),\nsecond-order derivatives\n(like d\n2\ny\/dx\n2\n), etc in ordinary derivatives. Likewise, we have first-order, second-order, and higher-order derivatives in partial derivatives also.\nFirst Order Partial Derivatives\nIf z = f(x, y) is a function in two variables, then it can have two first-order partial derivatives, namely ∂f \/ ∂x and ∂f \/ ∂y.\nExample:\nIf z = x\n2\n+ y\n2\n, find all the first order partial derivatives.\nSolution:\nf\nx\n= ∂f \/ ∂x = ∂ \/ ∂x (x\n2\n+ y\n2\n)\n= ∂ \/ ∂x (x\n2\n) + ∂ \/ ∂x (y\n2\n)\n= 2x + 0 (as y is a constant)\n= 2x\nf\ny\n= ∂f \/ ∂y = ∂ \/ ∂y (x\n2\n+ y\n2\n)\n= ∂ \/ ∂y (x\n2\n) + ∂ \/ ∂y (y\n2\n)\n= 0 + 2y (as x is a constant)\n= 2y\nSecond Order Partial Derivatives\nThe second-order partial derivative is obtained by differentiating the function with respect to the indicated variables successively one after the other. If z = f(x, y) is a function in two variables, then it can have four second-order partial derivatives, namely ∂\n2\nf \/ ∂x\n2\n, ∂\n2\nf \/ ∂y\n2\n, ∂\n2\nf \/ ∂x ∂y and ∂\n2\nf \/ ∂y ∂x. To find them, we can first differentiate the function partially with the latter variable, and then partially differentiate the result with respect to the former variable. i.e.,\nf\nxx\n= ∂\n2\nf \/ ∂x\n2\n= ∂ \/ ∂x (∂f \/ ∂x) = ∂ \/ ∂x (f\nx\n)\nf\nyy\n= ∂\n2\nf \/ ∂y\n2\n= ∂ \/ ∂y (∂f \/ ∂y) = ∂ \/ ∂x (f\ny\n)\nf\nyx\n= ∂\n2\nf \/ ∂x ∂y = ∂ \/ ∂x (∂f \/ ∂y) = ∂ \/ ∂x (f\ny\n)\nf\nxy\n= ∂\n2\nf \/ ∂y ∂x = ∂ \/ ∂y (∂f \/ ∂x) = ∂ \/ ∂y (f\nx\n)\nObserve the notations f\nyx\nand f\nxy\n. The order of variables in each subscript indicate the order of partial differentiation. For example, f\nyx\nmeans to partially differentiate with respect to y first and then with respect to x, and this is same as ∂\n2\nf \/ ∂x ∂y.\nExample:\nIf z = x\n2\n+ y\n2\n, find all the second order partial derivatives.\nSolution:\nIn the above example, we have already found that f\nx\n= 2x and f\ny\n= 2y.\nNow, f\nxx\n= ∂ \/ ∂x (f\nx\n) = ∂ \/ ∂x (2x) = 2\nf\nyy\n= ∂ \/ ∂y (f\ny\n) = ∂ \/ ∂y (2y) = 2\nf\nyx\n= ∂ \/ ∂x (f\ny\n) = ∂ \/ ∂x (2y) = 0\nf\nxy\n= ∂ \/ ∂y (f\nx\n) = ∂ \/ ∂y (2x) = 0\nNow that f\nyx\n= f\nxy\n. Thus, the order of partial differentiation doesn't matter.\nThe process of finding partial derivatives is known as Partial Differentiation. To find the first-order partial derivatives (as discussed earlier) of a function z = f(x, y) we use the following limit formulas:\n∂f \/ ∂x = lim\nh → 0\n[ f(x + h, y) - f(x, y) ] \/ h\n∂f \/ ∂y = lim\nh → 0\n[ f(x, y + h) - f(x, y) ] \/ h\nBut instead of using these formulas, just treating all the other variables than the variable with respect to which we are partially differentiating as constants would make the process of partial differentiation very easier. In this process, we just use the same\nrules as ordinary differentiation\nand among them, the important rules are as follows:\nPower Rule\nThe power rule of differentiation says d\/dx (x\nn\n) = n x\nn-1\n. The same rule can be applied in partial derivatives also.\nExample:\n∂ \/ ∂x (x\n2\ny) = y ∂ \/ ∂x (x\n2\n) = y (2x) = 2xy.\nProduct Rule\nThe\nproduct rule\nof ordinary differentiation says d\/dx (uv) = u dv\/dx + v du\/dx. We can apply the same rule in partial differentiation as well when there are two functions of the same variable.\nExample:\n∂ \/ ∂x (xy sin x) = y ∂ \/ ∂x (x sin x)\n= y [ x ∂ \/ ∂x (sin x) + sin x ∂ \/ ∂x (x) ]\n= y [ x cos x + sin x]\nQuotient Rule\nThe\nquotient rule\nof ordinary differentiation says d\/dx (u\/v) = [ v du\/dx - u dv\/dx ] \/ v\n2\n. As other rules, this rule can be applied for finding partial derivatives also.\nExample:\n∂ \/ ∂x ( xy \/ sin x)\n= y ∂ \/ ∂x (x \/ sin x)\n= y [ ( sin x ∂ \/ ∂x (x) - x ∂ \/ ∂x (sin x) ) \/ sin\n2\nx ]\n= y [ sin x - x cos x] \/ sin\n2\nx\nChain Rule of Partial Differentiation\nThe chain rule is used when we have to differentiate an\nimplicit function\n. The chain rule of partial derivatives works a little differently when compared to ordinary derivatives. Sometimes, the rule involves both partial derivatives and ordinary derivatives. There are various forms of this rule and the application of one of them depends upon how each variable of the function is defined.\nIf y = f(x) is a function where x is again a function of two variables u and v (i.e., x = x (u, v)) then\n∂f\/∂u = ∂f\/∂x · ∂x\/∂u;\n∂f\/∂v = ∂f\/∂x · ∂x\/∂v\nIf z = f(x, y), where each of x and y are again functions of a variable t (i.e., x = x(t) and y = y(t)) then\ndf\/dt = (∂f\/∂x · dx\/dt) + (∂f\/∂y · dy\/dt)\nIf z = f(x, y) is a function and each of x and y are again functions of two variables u and v (i.e., x = x(u, v) and y = y(u, v)) then\n∂f\/∂u = ∂f\/∂x · ∂x\/∂u + ∂f\/∂y · ∂y\/∂u;\n∂f\/∂v = ∂f\/∂x · ∂x\/∂v + ∂f\/∂y · ∂y\/∂v\nExample:\nIf z = e\nxy\n, where x = uv and y = u + v then find the partial derivative ∂f\/∂u.\nSolution:\nBy the chain rule of partial derivatives:\n∂f\/∂u = ∂f\/∂x · ∂x\/∂u + ∂f\/∂y · ∂y\/∂u\n= ∂ \/ ∂x (e\nxy\n) · ∂ \/ ∂u (uv) + ∂ \/ ∂y (e\nxy\n) · ∂ \/ ∂u (u + v)\n= (e\nxy\n· y) (v) + (e\nxy\n· x) (1)\n= e\nxy\n(x + vy)\nOther Rules of Partial Differentiation\nIf f(x, y) = a constant, then the following formula gives the relation between the ordinary derivative and the partial derivatives which follows from\nimplicit differentiation\n.\ndy\/dx = -f\nx\n\/f\ny\n.\nFor any two functions u(x, y) and v(x, y), the\ndeterminant\n|\n∂\nu\n\/\n∂\nx\n∂\nu\n\/\n∂\ny\n∂\nv\n\/\n∂\nx\n∂\nv\n\/\n∂\ny\n|\nis known as\nJacobian\nof u and v.\nThe\nLaplace equation\nof partial derivatives is ∂\n2\nf \/ ∂x\n2\n+ ∂\n2\nf \/ ∂y\n2\n+ ∂\n2\nf \/ ∂z\n2\n= 0 where f(x, y, z) is a function in three variables. Any function f that satisfies the Laplace equation is known as the harmonic function.\nImportant Notes on Partial Derivatives:\nWhile finding the partial derivative with respect to a variable, all the other variables should be considered as constants.\nThe order of taking derivatives doesn't matter in partial derivatives. i.e., ∂\n2\nf \/ ∂x ∂y = ∂\n2\nf \/ ∂y ∂x.\nThe rules of derivatives apply for partial differentiation as well.\nInstead of using the limit definition, applying\nderivative formulas\nmake the process of finding the partial derivatives easier.\n☛\nRelated Topics:\nPartial Differential Equations\nPartial Derivative Calculator\nRead More","attrs_markdown":"[Math Classes](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/)\n\n[Year K-2](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/year-1-3\/)\n\n[Kindergarten](https:\/\/www.cuemath.com\/en-eu\/math-online-classes\/kindergarten\/)\n\n[Year 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App](https:\/\/static.qumath.in\/static\/next\/_next\/static\/images\/app_banner_desktop-8753ba6fb9ffbf8a558f26e1450c864e.webp)\n\n![Download the Cuemath App](https:\/\/static.qumath.in\/static\/next\/_next\/static\/images\/app_banner_mobile-3f26e4236459d2a093f12eabcd31dc32.webp)\n\n![Download the Cuemath App](https:\/\/static.qumath.in\/static\/next\/_next\/static\/images\/app_banner_desktop-8753ba6fb9ffbf8a558f26e1450c864e.webp)\n\n![Download the Cuemath App](https:\/\/static.qumath.in\/static\/next\/_next\/static\/images\/app_banner_mobile-3f26e4236459d2a093f12eabcd31dc32.webp)\n\n[Learn]()\n\n[Practice]()[Download]()\n\n# Partial Derivative\nThe partial derivative of a function (in two or more variables) is its derivative with respect to one of the variables keeping all the other variables as constants. The process of calculating partial derivative is as same as that of an ordinary derivative except we consider the other variables than the variable with respect to which we are differentiating as constants.\n\nLet us learn more about how to calculate partial derivatives of different orders along with examples.\n\n|   |   |\n|---|---|\n| 1\\. | [What is Partial Derivative?](https:\/\/www.cuemath.com\/calculus\/partial-derivative\/#a) |\n| 2\\. | [Partial Derivative Symbol](https:\/\/www.cuemath.com\/calculus\/partial-derivative\/#b) |\n| 3\\. | [How to Calculate Partial Derivatives?](https:\/\/www.cuemath.com\/calculus\/partial-derivative\/#c) |\n| 4\\. | [Partial Derivatives of Different Orders](https:\/\/www.cuemath.com\/calculus\/partial-derivative\/#d) |\n| 5\\. | [Partial Differentiation Formulas](https:\/\/www.cuemath.com\/calculus\/partial-derivative\/#e) |\n| 6\\. | [FAQs on Partial Derivative](https:\/\/www.cuemath.com\/calculus\/partial-derivative\/#faqs) |\n## What is Partial Derivative?\nThe **partial derivative** of a multivariable function, say z = f(x, y), is its [derivative](https:\/\/www.cuemath.com\/calculus\/differentiable\/) with respect to one of the variables, x or y in this case, where the other variables are treated as constants. For example,\n\n- for finding the partial derivative of f(x, y) with respect to x (which is represented by ∂f \/ ∂x), y is treated as constant and\n- for finding the partial derivative of f(x, y) with respect to y (which is represented by ∂f \/ ∂y), x is treated as constant\n\nNote that we are not considering all the variables as variables while doing partial differentiation (instead, we are considering only one variable as a variable at a time) and hence the name \"partial\". The limit definition of a partial derivative looks very similar to the [limit](https:\/\/www.cuemath.com\/calculus\/limits\/) definition of the derivative. We can find the partial derivatives using the following limit formulas:\n\n- ∂f \/ ∂x = lim h → 0 \\[ f(x + h, y) - f(x, y) \\] \/ h\n- ∂f \/ ∂y = lim h → 0 \\[ f(x, y + h) - f(x, y) \\] \/ h\n\n![partial derivative formula](https:\/\/d138zd1ktt9iqe.cloudfront.net\/media\/seo_landing_files\/partial-derivative-formula-1648621191.png)\n\nThese formulas resemble the derivative definition using the first principle.\n\n### Example of Partial Derivative\nIf f (x, y) = xy, then find the partial derivative ∂f \/ ∂x.\n\n**Solution:**\n\n∂f \/ ∂x = lim h → 0 \\[ f(x + h, y) - f(x, y) \\] \/ h  \n = lim h → 0 \\[ (x + h)y - xy \\] \/ h  \n = lim h → 0 \\[xy + hy - xy\\] \/ h  \n = lim h → 0 \\[hy\\]\/h  \n = lim h → 0 y  \n = y\n\nTherefore, ∂f \/ ∂x = y.\n\n## Partial Derivative Symbol\nWe know that the ordinary derivative of a function y = f(x) is denoted by one of the notations dy\/dx, d\/dx (y), d\/dx (f(x)), f '(x), etc. For representing a partial derivative we use the symbol \"∂\" instead of \"d\". We pronounce \"∂\" to be \"doh\" but it has some other names like \"partial\", \"del\", \"partial dee\", \"dee\", \"Jacobi's delta\", etc. If z = f(x, y) is a function in two variables then\n\n- ∂f \/ ∂x is the partial derivative of f with respect to x\n- ∂f \/ ∂y is the partial derivative of f with respect to y\n\nJust like how we have different [symbols](https:\/\/www.cuemath.com\/numbers\/math-symbols\/) of ordinary derivatives, we have different notations for partial derivatives as well. For example, ∂f \/ ∂x can be written as fx, fx', Dxf, ∂ \/ ∂x (f), ∂x f, ∂ \/ ∂x \\[f(x, y)\\], ∂z \/ ∂x, etc.\n\n## Calculate Partial Derivatives\nWe have already seen that the limit definitions are used to find the partial derivatives. But using the [limit formula](https:\/\/www.cuemath.com\/limit-formula\/) and computing the limit is not always easy. Thus, we have another method to calculate partial derivatives that follow right from its definition. In this method, if z = f(x, y) is the function, then we can compute the partial derivatives using the following steps:\n\n- **Step 1**: Identify the variable with respect to which we have to find the partial derivative.\n- **Step 2:** Except for the variable found in **Step 1**, treat all the other variables as constants.\n- **Step 3:** Differentiate the function just using the rules of ordinary differentiation.\n\nWait! Read **Step 3** again. Yes, the rules of ordinary differentiation are as same as that of partial differentiation. In partial differentiation, just treating variables is different, that's it\\!\n\n**Example:** Let us solve the same above example (If f (x, y) = xy, then find the partial derivative ∂f \/ ∂x) using the above steps.\n\n**Solution:**\n\nWe have to find ∂f \/ ∂x. It means, we have to find the partial derivative of f with respect to x. So we treat y as constant. Thus, we can write 'y' outside the derivative (as in ordinary differentiation, we have a rule that says d\/dx (c y) = c dy\/dx, where 'c' is a constant). Thus,\n\n∂f \/ ∂x = ∂ \/ ∂x (xy)  \n = y ∂ \/ ∂x (x)  \n = y (1) (Using [power rule](https:\/\/www.cuemath.com\/calculus\/power-rule\/), d\/dx (x) = 1)  \n = y\n\nWe have got the same answer as we got using the limit definition.\n\n## Partial Derivatives of Different Orders\nWe have derivatives like first-order derivatives (like dy\/dx), [second-order derivatives](https:\/\/www.cuemath.com\/calculators\/second-derivative-calculator\/) (like d2y\/dx2), etc in ordinary derivatives. Likewise, we have first-order, second-order, and higher-order derivatives in partial derivatives also.\n\n### First Order Partial Derivatives\nIf z = f(x, y) is a function in two variables, then it can have two first-order partial derivatives, namely ∂f \/ ∂x and ∂f \/ ∂y.\n\n**Example:** If z = x2 + y2, find all the first order partial derivatives.\n\n**Solution:**\n\nfx = ∂f \/ ∂x = ∂ \/ ∂x (x2 + y2)  \n = ∂ \/ ∂x (x2) + ∂ \/ ∂x (y2)  \n = 2x + 0 (as y is a constant)  \n = 2x\n\nfy = ∂f \/ ∂y = ∂ \/ ∂y (x2 + y2)  \n = ∂ \/ ∂y (x2) + ∂ \/ ∂y (y2)  \n = 0 + 2y (as x is a constant)  \n = 2y\n\n### Second Order Partial Derivatives\nThe second-order partial derivative is obtained by differentiating the function with respect to the indicated variables successively one after the other. If z = f(x, y) is a function in two variables, then it can have four second-order partial derivatives, namely ∂2f \/ ∂x2, ∂2f \/ ∂y2, ∂2f \/ ∂x ∂y and ∂2f \/ ∂y ∂x. To find them, we can first differentiate the function partially with the latter variable, and then partially differentiate the result with respect to the former variable. i.e.,\n\n- fxx = ∂2f \/ ∂x2 = ∂ \/ ∂x (∂f \/ ∂x) = ∂ \/ ∂x (fx)\n- fyy = ∂2f \/ ∂y2 = ∂ \/ ∂y (∂f \/ ∂y) = ∂ \/ ∂x (fy)\n- fyx = ∂2f \/ ∂x ∂y = ∂ \/ ∂x (∂f \/ ∂y) = ∂ \/ ∂x (fy)\n- fxy = ∂2f \/ ∂y ∂x = ∂ \/ ∂y (∂f \/ ∂x) = ∂ \/ ∂y (fx)\n\nObserve the notations fyx and fxy. The order of variables in each subscript indicate the order of partial differentiation. For example, fyx means to partially differentiate with respect to y first and then with respect to x, and this is same as ∂2f \/ ∂x ∂y.\n\n**Example:** If z = x2 + y2, find all the second order partial derivatives.\n\n**Solution:**\n\nIn the above example, we have already found that fx = 2x and fy = 2y.\n\nNow, fxx = ∂ \/ ∂x (fx) = ∂ \/ ∂x (2x) = 2\n\nfyy = ∂ \/ ∂y (fy) = ∂ \/ ∂y (2y) = 2\n\nfyx = ∂ \/ ∂x (fy) = ∂ \/ ∂x (2y) = 0\n\nfxy = ∂ \/ ∂y (fx) = ∂ \/ ∂y (2x) = 0\n\nNow that fyx = fxy. Thus, the order of partial differentiation doesn't matter.\n\n## Partial Differentiation Formulas\nThe process of finding partial derivatives is known as Partial Differentiation. To find the first-order partial derivatives (as discussed earlier) of a function z = f(x, y) we use the following limit formulas:\n\n- ∂f \/ ∂x = lim h → 0 \\[ f(x + h, y) - f(x, y) \\] \/ h\n- ∂f \/ ∂y = lim h → 0 \\[ f(x, y + h) - f(x, y) \\] \/ h\n\nBut instead of using these formulas, just treating all the other variables than the variable with respect to which we are partially differentiating as constants would make the process of partial differentiation very easier. In this process, we just use the same [rules as ordinary differentiation](https:\/\/www.cuemath.com\/calculus\/derivative-rules\/) and among them, the important rules are as follows:\n\n### Power Rule\nThe power rule of differentiation says d\/dx (xn) = n xn-1. The same rule can be applied in partial derivatives also.\n\n**Example:** ∂ \/ ∂x (x2y) = y ∂ \/ ∂x (x2) = y (2x) = 2xy.\n\n### Product Rule\nThe [product rule](https:\/\/www.cuemath.com\/calculus\/product-rule\/) of ordinary differentiation says d\/dx (uv) = u dv\/dx + v du\/dx. We can apply the same rule in partial differentiation as well when there are two functions of the same variable.\n\n**Example:** ∂ \/ ∂x (xy sin x) = y ∂ \/ ∂x (x sin x)  \n = y \\[ x ∂ \/ ∂x (sin x) + sin x ∂ \/ ∂x (x) \\]  \n = y \\[ x cos x + sin x\\]\n\n### Quotient Rule\nThe [quotient rule](https:\/\/www.cuemath.com\/calculus\/quotient-rule\/) of ordinary differentiation says d\/dx (u\/v) = \\[ v du\/dx - u dv\/dx \\] \/ v2. As other rules, this rule can be applied for finding partial derivatives also.\n\n**Example:** ∂ \/ ∂x ( xy \/ sin x)  \n = y ∂ \/ ∂x (x \/ sin x)  \n = y \\[ ( sin x ∂ \/ ∂x (x) - x ∂ \/ ∂x (sin x) ) \/ sin2x \\]  \n = y \\[ sin x - x cos x\\] \/ sin2x\n\n### Chain Rule of Partial Differentiation\nThe chain rule is used when we have to differentiate an [implicit function](https:\/\/www.cuemath.com\/algebra\/implicit-function\/). The chain rule of partial derivatives works a little differently when compared to ordinary derivatives. Sometimes, the rule involves both partial derivatives and ordinary derivatives. There are various forms of this rule and the application of one of them depends upon how each variable of the function is defined.\n\n- If y = f(x) is a function where x is again a function of two variables u and v (i.e., x = x (u, v)) then  \n   ∂f\/∂u = ∂f\/∂x · ∂x\/∂u;  \n   ∂f\/∂v = ∂f\/∂x · ∂x\/∂v\n- If z = f(x, y), where each of x and y are again functions of a variable t (i.e., x = x(t) and y = y(t)) then  \n   df\/dt = (∂f\/∂x · dx\/dt) + (∂f\/∂y · dy\/dt)\n- If z = f(x, y) is a function and each of x and y are again functions of two variables u and v (i.e., x = x(u, v) and y = y(u, v)) then  \n   ∂f\/∂u = ∂f\/∂x · ∂x\/∂u + ∂f\/∂y · ∂y\/∂u;  \n   ∂f\/∂v = ∂f\/∂x · ∂x\/∂v + ∂f\/∂y · ∂y\/∂v\n\n**Example:** If z = exy, where x = uv and y = u + v then find the partial derivative ∂f\/∂u.\n\n**Solution:**\n\nBy the chain rule of partial derivatives:\n\n∂f\/∂u = ∂f\/∂x · ∂x\/∂u + ∂f\/∂y · ∂y\/∂u  \n = ∂ \/ ∂x (exy) · ∂ \/ ∂u (uv) + ∂ \/ ∂y (exy) · ∂ \/ ∂u (u + v)  \n = (exy · y) (v) + (exy · x) (1)  \n = exy (x + vy)\n\n### Other Rules of Partial Differentiation\n- If f(x, y) = a constant, then the following formula gives the relation between the ordinary derivative and the partial derivatives which follows from [implicit differentiation](https:\/\/www.cuemath.com\/calculus\/implicit-differentiation\/).  \n   dy\/dx = -fx\/fy.\n- For any two functions u(x, y) and v(x, y), the [determinant](https:\/\/www.cuemath.com\/algebra\/determinant-of-matrix\/)\n  ∣∣∣∂u\/∂x∂u\/∂y∂v\/∂x∂v\/∂y∣∣∣\n  \n  \\|\n  \n  ∂\n  \n  u\n  \n  \/\n  \n  ∂\n  \n  x\n  \n  ∂\n  \n  u\n  \n  \/\n  \n  ∂\n  \n  y\n  \n  ∂\n  \n  v\n  \n  \/\n  \n  ∂\n  \n  x\n  \n  ∂\n  \n  v\n  \n  \/\n  \n  ∂\n  \n  y\n  \n  \\|\n  is known as **Jacobian** of u and v.\n- The **Laplace equation** of partial derivatives is ∂2f \/ ∂x2 + ∂2f \/ ∂y2 + ∂2f \/ ∂z2 = 0 where f(x, y, z) is a function in three variables. Any function f that satisfies the Laplace equation is known as the harmonic function.\n\n**Important Notes on Partial Derivatives:**\n\n- While finding the partial derivative with respect to a variable, all the other variables should be considered as constants.\n- The order of taking derivatives doesn't matter in partial derivatives. i.e., ∂2f \/ ∂x ∂y = ∂2f \/ ∂y ∂x.\n- The rules of derivatives apply for partial differentiation as well.\n- Instead of using the limit definition, applying [derivative formulas](https:\/\/www.cuemath.com\/derivative-formula\/) make the process of finding the partial derivatives easier.\n\n☛ **Related Topics:**\n\n- [Partial Differential Equations](https:\/\/www.cuemath.com\/calculus\/partial-differential-equations\/)\n- [Partial Derivative Calculator](https:\/\/www.cuemath.com\/calculators\/partial-derivative-calculator\/)\n\nRead More\n\nExplore  \n math program\n\nDownload FREE Study Materials\n\n![](data:image\/svg+xml;base64,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)\n\nSHEETS\n\n![](https:\/\/d138zd1ktt9iqe.cloudfront.net\/media\/seo_landing_files\/thumbnails\/partial-derivative-worksheet-newui-first-page-1649086216.jpg)\n\nPartial Derivative Worksheet\n\nCalculus Worksheet\n\n![](data:image\/svg+xml;base64,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)\n\n## Partial Derivative Examples\n1. **Example 1:** Find all the first order partial derivatives of the function f(x, y) = ax2 + 2hxy + by2.\n   \n   **Solution:**\n   \n   The first-order partial derivatives are:\n   \n   fx = ∂f \/ ∂x = ∂ \/ ∂x (ax2 + 2hxy + by2)  \n    = ∂ \/ ∂x (ax2) + ∂ \/ ∂x (2hxy) + ∂ \/ ∂x (by2)  \n    = 2ax + 2hy (1) + 0 (as y is a constant)  \n    = 2ax + 2hy\n   \n   fx = ∂f \/ ∂x = ∂ \/ ∂x (ax2 + 2hxy + by2)  \n    = ∂ \/ ∂y (ax2) + ∂ \/ ∂y (2hxy) + ∂ \/ ∂y (by2)  \n    = 0 + 2hx (1) + 2by (as x is a constant)  \n    = 2hx + 2by\n   \n   **Answer:** fx = 2ax + 2hy and fy = 2hx + 2by.\n2. **Example 2:** Find all the second-order partial derivatives of the function given in **Example 1.**\n   \n   **Solution:**\n   \n   We found in **Example 1** thatfx = 2ax + 2hy and fy = 2hx + 2by. Now, we will find the second-order partial derivatives.\n   \n   fxx = ∂ \/ ∂x (fx) = ∂ \/ ∂x (2ax + 2hy) = 2a\n   \n   fyy = ∂ \/ ∂y (fy) = ∂ \/ ∂y (2hx + 2by) = 2b\n   \n   fyx = ∂ \/ ∂x (fy) = ∂ \/ ∂x (2hx + 2by) = 2h\n   \n   fxy = ∂ \/ ∂y (fx) = ∂ \/ ∂y (2ax + 2hy) = 2h\n   \n   **Answer:** We found that fxx = 2a, fyy = 2b, fyx = 2h, and fxy = 2h.\n3. **Example 3:** Does the order of differentiation matter while finding the partial derivatives? Justify your answer using **Example 2.**\n   \n   **Solution:**\n   \n   From **Example 2**, fyx = 2h and fxy = 2h.\n   \n   Thus, fxy = fxy.\n   \n   Thus, the order of partial differentiation doesn't matter.\n   \n   **Answer:** The order doesn't matter as fyx = fxy.\n\nView Answer \\>\n\n![](https:\/\/d138zd1ktt9iqe.cloudfront.net\/media\/seo_landing_files\/caucasian-kid-and-indian-teacher-grade-4-7-1619081163.png)\n\nGreat learning in high school using simple cues\n\nIndulging in rote learning, you are likely to forget concepts. With Cuemath, you will learn visually and be surprised by the outcomes.\n\n[Book a Free Trial Class](https:\/\/www.cuemath.com\/parent\/signup\/)\n\n## Practice Questions on Partial Differentiation\nCheck Answer \\>\n\n## FAQs on Partial Derivative\n### How to Calculate Partial Derivatives?\nWe use **partial derivatives** when the function has more than one variable. If a function f is in terms of two variables x and y, then we can calculate the partial derivatives as follows.\n\n- the partial derivative of f = ∂f\/∂x and y has to be treated as constant here.\n- the partial derivative of f = ∂f\/∂y and x has to be treated as constant here.\n\n### What is the Symbol of Partial Derivatives?\nWe use the symbol ∂ to represent a partial derivative. For example, the partial derivative of a function f(x, y) with respect to x is written as ∂f\/∂x.\n\n### What is the Difference Between Differentiation and Partial Differentiation?\nWe talk about the [derivative](https:\/\/www.cuemath.com\/calculus\/derivatives\/) of a function if a function has only one variable in it. For example, the derivative of a function y = f(x) is denoted by df\/dx. We talk about partial derivatives when a function z = f(x, y) has more than one variable. The partial derivative of f with respect to x is denoted by ∂f\/∂x and while finding this, we treat y as constant.\n\n### What is the Formula Used to Find the Partial Derivative?\nThe partial derivatives of a function z = f(x, y) can be found using the limit formulas:\n\n- ∂f \/ ∂x = lim h → 0 \\[ f(x + h, y) - f(x, y) \\] \/ h\n- ∂f \/ ∂y = lim h → 0 \\[ f(x, y + h) - f(x, y) \\] \/ h\n\n### What Does Partial Derivative Tell Us?\nPartial derivative tells us to [differentiate](https:\/\/www.cuemath.com\/calculus\/differentiation\/) a function partially. It means if we are differentiating partially with respect to one variable, then the remaining [variables](https:\/\/www.cuemath.com\/algebra\/variables-constants-and-expressions\/) of the function must be treated as constants.\n\n### What is the Chain Rule of Partial Derivatives?\nThe chain rule of partial derivative is mentioned below: If z = f(x, y) is a [function](https:\/\/www.cuemath.com\/calculus\/What-are-functions\/) where x and y are functions of two variables u and v (i.e., x = x(u, v) and y = y(u, v)) then by the chain rule of partial derivatives,\n\n- ∂f\/∂u = ∂f\/∂x · ∂x\/∂u + ∂f\/∂y · ∂y\/∂u\n- ∂f\/∂v = ∂f\/∂x · ∂x\/∂v + ∂f\/∂y · ∂y\/∂v\n\n### What is the Formula that Connects Normal Derivatives with Partial Derivatives?\nWhen f(x, y) = c, where 'c' is a [constant](https:\/\/www.cuemath.com\/algebra\/constants\/), then dy\/dx = -fx\/fy, where\n\n- fx is the partial derivative of f with respect to x\n- fy is the partial derivative of f with respect to y\n\nExplore  \n math 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The process of calculating partial derivative is as same as that of an ordinary derivative except we consider the other variables than the variable with respect to which we are differentiating as constants.\n\nLet us learn more about how to calculate partial derivatives of different orders along with examples.\n\n|   |   |\n|---|---|\n| 1\\. | [What is Partial Derivative?](https:\/\/www.cuemath.com\/calculus\/partial-derivative\/#a) |\n| 2\\. | [Partial Derivative Symbol](https:\/\/www.cuemath.com\/calculus\/partial-derivative\/#b) |\n| 3\\. | [How to Calculate Partial Derivatives?](https:\/\/www.cuemath.com\/calculus\/partial-derivative\/#c) |\n| 4\\. | [Partial Derivatives of Different Orders](https:\/\/www.cuemath.com\/calculus\/partial-derivative\/#d) |\n| 5\\. | [Partial Differentiation Formulas](https:\/\/www.cuemath.com\/calculus\/partial-derivative\/#e) |\n| 6\\. | [FAQs on Partial Derivative](https:\/\/www.cuemath.com\/calculus\/partial-derivative\/#faqs) |\n\nThe **partial derivative** of a multivariable function, say z = f(x, y), is its [derivative](https:\/\/www.cuemath.com\/calculus\/differentiable\/) with respect to one of the variables, x or y in this case, where the other variables are treated as constants. For example,\n\n- for finding the partial derivative of f(x, y) with respect to x (which is represented by ∂f \/ ∂x), y is treated as constant and\n- for finding the partial derivative of f(x, y) with respect to y (which is represented by ∂f \/ ∂y), x is treated as constant\n\nNote that we are not considering all the variables as variables while doing partial differentiation (instead, we are considering only one variable as a variable at a time) and hence the name \"partial\". The limit definition of a partial derivative looks very similar to the [limit](https:\/\/www.cuemath.com\/calculus\/limits\/) definition of the derivative. We can find the partial derivatives using the following limit formulas:\n\n- ∂f \/ ∂x = lim h → 0 \\[ f(x + h, y) - f(x, y) \\] \/ h\n- ∂f \/ ∂y = lim h → 0 \\[ f(x, y + h) - f(x, y) \\] \/ h\n\nThese formulas resemble the derivative definition using the first principle.\n\n### Example of Partial Derivative\nIf f (x, y) = xy, then find the partial derivative ∂f \/ ∂x.\n\n**Solution:**\n\n∂f \/ ∂x = lim h → 0 \\[ f(x + h, y) - f(x, y) \\] \/ h  \n = lim h → 0 \\[ (x + h)y - xy \\] \/ h  \n = lim h → 0 \\[xy + hy - xy\\] \/ h  \n = lim h → 0 \\[hy\\]\/h  \n = lim h → 0 y  \n = y\n\nTherefore, ∂f \/ ∂x = y.\n\nWe know that the ordinary derivative of a function y = f(x) is denoted by one of the notations dy\/dx, d\/dx (y), d\/dx (f(x)), f '(x), etc. For representing a partial derivative we use the symbol \"∂\" instead of \"d\". We pronounce \"∂\" to be \"doh\" but it has some other names like \"partial\", \"del\", \"partial dee\", \"dee\", \"Jacobi's delta\", etc. If z = f(x, y) is a function in two variables then\n\n- ∂f \/ ∂x is the partial derivative of f with respect to x\n- ∂f \/ ∂y is the partial derivative of f with respect to y\n\nJust like how we have different [symbols](https:\/\/www.cuemath.com\/numbers\/math-symbols\/) of ordinary derivatives, we have different notations for partial derivatives as well. For example, ∂f \/ ∂x can be written as fx, fx', Dxf, ∂ \/ ∂x (f), ∂x f, ∂ \/ ∂x \\[f(x, y)\\], ∂z \/ ∂x, etc.\n\nWe have already seen that the limit definitions are used to find the partial derivatives. But using the [limit formula](https:\/\/www.cuemath.com\/limit-formula\/) and computing the limit is not always easy. Thus, we have another method to calculate partial derivatives that follow right from its definition. In this method, if z = f(x, y) is the function, then we can compute the partial derivatives using the following steps:\n\n- **Step 1**: Identify the variable with respect to which we have to find the partial derivative.\n- **Step 2:** Except for the variable found in **Step 1**, treat all the other variables as constants.\n- **Step 3:** Differentiate the function just using the rules of ordinary differentiation.\n\nWait! Read **Step 3** again. Yes, the rules of ordinary differentiation are as same as that of partial differentiation. In partial differentiation, just treating variables is different, that's it\\!\n\n**Example:** Let us solve the same above example (If f (x, y) = xy, then find the partial derivative ∂f \/ ∂x) using the above steps.\n\n**Solution:**\n\nWe have to find ∂f \/ ∂x. It means, we have to find the partial derivative of f with respect to x. So we treat y as constant. Thus, we can write 'y' outside the derivative (as in ordinary differentiation, we have a rule that says d\/dx (c y) = c dy\/dx, where 'c' is a constant). Thus,\n\n∂f \/ ∂x = ∂ \/ ∂x (xy)  \n = y ∂ \/ ∂x (x)  \n = y (1) (Using [power rule](https:\/\/www.cuemath.com\/calculus\/power-rule\/), d\/dx (x) = 1)  \n = y\n\nWe have got the same answer as we got using the limit definition.\n\nWe have derivatives like first-order derivatives (like dy\/dx), [second-order derivatives](https:\/\/www.cuemath.com\/calculators\/second-derivative-calculator\/) (like d2y\/dx2), etc in ordinary derivatives. Likewise, we have first-order, second-order, and higher-order derivatives in partial derivatives also.\n\n### First Order Partial Derivatives\nIf z = f(x, y) is a function in two variables, then it can have two first-order partial derivatives, namely ∂f \/ ∂x and ∂f \/ ∂y.\n\n**Example:** If z = x2 + y2, find all the first order partial derivatives.\n\n**Solution:**\n\nfx = ∂f \/ ∂x = ∂ \/ ∂x (x2 + y2)  \n = ∂ \/ ∂x (x2) + ∂ \/ ∂x (y2)  \n = 2x + 0 (as y is a constant)  \n = 2x\n\nfy = ∂f \/ ∂y = ∂ \/ ∂y (x2 + y2)  \n = ∂ \/ ∂y (x2) + ∂ \/ ∂y (y2)  \n = 0 + 2y (as x is a constant)  \n = 2y\n\n### Second Order Partial Derivatives\nThe second-order partial derivative is obtained by differentiating the function with respect to the indicated variables successively one after the other. If z = f(x, y) is a function in two variables, then it can have four second-order partial derivatives, namely ∂2f \/ ∂x2, ∂2f \/ ∂y2, ∂2f \/ ∂x ∂y and ∂2f \/ ∂y ∂x. To find them, we can first differentiate the function partially with the latter variable, and then partially differentiate the result with respect to the former variable. i.e.,\n\n- fxx = ∂2f \/ ∂x2 = ∂ \/ ∂x (∂f \/ ∂x) = ∂ \/ ∂x (fx)\n- fyy = ∂2f \/ ∂y2 = ∂ \/ ∂y (∂f \/ ∂y) = ∂ \/ ∂x (fy)\n- fyx = ∂2f \/ ∂x ∂y = ∂ \/ ∂x (∂f \/ ∂y) = ∂ \/ ∂x (fy)\n- fxy = ∂2f \/ ∂y ∂x = ∂ \/ ∂y (∂f \/ ∂x) = ∂ \/ ∂y (fx)\n\nObserve the notations fyx and fxy. The order of variables in each subscript indicate the order of partial differentiation. For example, fyx means to partially differentiate with respect to y first and then with respect to x, and this is same as ∂2f \/ ∂x ∂y.\n\n**Example:** If z = x2 + y2, find all the second order partial derivatives.\n\n**Solution:**\n\nIn the above example, we have already found that fx = 2x and fy = 2y.\n\nNow, fxx = ∂ \/ ∂x (fx) = ∂ \/ ∂x (2x) = 2\n\nfyy = ∂ \/ ∂y (fy) = ∂ \/ ∂y (2y) = 2\n\nfyx = ∂ \/ ∂x (fy) = ∂ \/ ∂x (2y) = 0\n\nfxy = ∂ \/ ∂y (fx) = ∂ \/ ∂y (2x) = 0\n\nNow that fyx = fxy. Thus, the order of partial differentiation doesn't matter.\n\nThe process of finding partial derivatives is known as Partial Differentiation. To find the first-order partial derivatives (as discussed earlier) of a function z = f(x, y) we use the following limit formulas:\n\n- ∂f \/ ∂x = lim h → 0 \\[ f(x + h, y) - f(x, y) \\] \/ h\n- ∂f \/ ∂y = lim h → 0 \\[ f(x, y + h) - f(x, y) \\] \/ h\n\nBut instead of using these formulas, just treating all the other variables than the variable with respect to which we are partially differentiating as constants would make the process of partial differentiation very easier. In this process, we just use the same [rules as ordinary differentiation](https:\/\/www.cuemath.com\/calculus\/derivative-rules\/) and among them, the important rules are as follows:\n\n### Power Rule\nThe power rule of differentiation says d\/dx (xn) = n xn-1. The same rule can be applied in partial derivatives also.\n\n**Example:** ∂ \/ ∂x (x2y) = y ∂ \/ ∂x (x2) = y (2x) = 2xy.\n\n### Product Rule\nThe [product rule](https:\/\/www.cuemath.com\/calculus\/product-rule\/) of ordinary differentiation says d\/dx (uv) = u dv\/dx + v du\/dx. We can apply the same rule in partial differentiation as well when there are two functions of the same variable.\n\n**Example:** ∂ \/ ∂x (xy sin x) = y ∂ \/ ∂x (x sin x)  \n = y \\[ x ∂ \/ ∂x (sin x) + sin x ∂ \/ ∂x (x) \\]  \n = y \\[ x cos x + sin x\\]\n\n### Quotient Rule\nThe [quotient rule](https:\/\/www.cuemath.com\/calculus\/quotient-rule\/) of ordinary differentiation says d\/dx (u\/v) = \\[ v du\/dx - u dv\/dx \\] \/ v2. As other rules, this rule can be applied for finding partial derivatives also.\n\n**Example:** ∂ \/ ∂x ( xy \/ sin x)  \n = y ∂ \/ ∂x (x \/ sin x)  \n = y \\[ ( sin x ∂ \/ ∂x (x) - x ∂ \/ ∂x (sin x) ) \/ sin2x \\]  \n = y \\[ sin x - x cos x\\] \/ sin2x\n\n### Chain Rule of Partial Differentiation\nThe chain rule is used when we have to differentiate an [implicit function](https:\/\/www.cuemath.com\/algebra\/implicit-function\/). The chain rule of partial derivatives works a little differently when compared to ordinary derivatives. Sometimes, the rule involves both partial derivatives and ordinary derivatives. There are various forms of this rule and the application of one of them depends upon how each variable of the function is defined.\n\n- If y = f(x) is a function where x is again a function of two variables u and v (i.e., x = x (u, v)) then  \n   ∂f\/∂u = ∂f\/∂x · ∂x\/∂u;  \n   ∂f\/∂v = ∂f\/∂x · ∂x\/∂v\n- If z = f(x, y), where each of x and y are again functions of a variable t (i.e., x = x(t) and y = y(t)) then  \n   df\/dt = (∂f\/∂x · dx\/dt) + (∂f\/∂y · dy\/dt)\n- If z = f(x, y) is a function and each of x and y are again functions of two variables u and v (i.e., x = x(u, v) and y = y(u, v)) then  \n   ∂f\/∂u = ∂f\/∂x · ∂x\/∂u + ∂f\/∂y · ∂y\/∂u;  \n   ∂f\/∂v = ∂f\/∂x · ∂x\/∂v + ∂f\/∂y · ∂y\/∂v\n\n**Example:** If z = exy, where x = uv and y = u + v then find the partial derivative ∂f\/∂u.\n\n**Solution:**\n\nBy the chain rule of partial derivatives:\n\n∂f\/∂u = ∂f\/∂x · ∂x\/∂u + ∂f\/∂y · ∂y\/∂u  \n = ∂ \/ ∂x (exy) · ∂ \/ ∂u (uv) + ∂ \/ ∂y (exy) · ∂ \/ ∂u (u + v)  \n = (exy · y) (v) + (exy · x) (1)  \n = exy (x + vy)\n\n### Other Rules of Partial Differentiation\n- If f(x, y) = a constant, then the following formula gives the relation between the ordinary derivative and the partial derivatives which follows from [implicit differentiation](https:\/\/www.cuemath.com\/calculus\/implicit-differentiation\/).  \n   dy\/dx = -fx\/fy.\n- For any two functions u(x, y) and v(x, y), the [determinant](https:\/\/www.cuemath.com\/algebra\/determinant-of-matrix\/)\n  \\|\n  \n  ∂\n  \n  u\n  \n  \/\n  \n  ∂\n  \n  x\n  \n  ∂\n  \n  u\n  \n  \/\n  \n  ∂\n  \n  y\n  \n  ∂\n  \n  v\n  \n  \/\n  \n  ∂\n  \n  x\n  \n  ∂\n  \n  v\n  \n  \/\n  \n  ∂\n  \n  y\n  \n  \\|\n  is known as **Jacobian** of u and v.\n- The **Laplace equation** of partial derivatives is ∂2f \/ ∂x2 + ∂2f \/ ∂y2 + ∂2f \/ ∂z2 = 0 where f(x, y, z) is a function in three variables. Any function f that satisfies the Laplace equation is known as the harmonic function.\n\n**Important Notes on Partial Derivatives:**\n\n- While finding the partial derivative with respect to a variable, all the other variables should be considered as constants.\n- The order of taking derivatives doesn't matter in partial derivatives. i.e., ∂2f \/ ∂x ∂y = ∂2f \/ ∂y ∂x.\n- The rules of derivatives apply for partial differentiation as well.\n- Instead of using the limit definition, applying [derivative formulas](https:\/\/www.cuemath.com\/derivative-formula\/) make the process of finding the partial derivatives easier.\n\n☛ **Related Topics:**\n\n- [Partial Differential Equations](https:\/\/www.cuemath.com\/calculus\/partial-differential-equations\/)\n- [Partial Derivative Calculator](https:\/\/www.cuemath.com\/calculators\/partial-derivative-calculator\/)\n\nRead More","meta_canonical":null}

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URL	https://www.cuemath.com/calculus/partial-derivative/
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Meta Title	Partial Derivative (Partial Differentiation) - Calculate, Symbol
Meta Description	The partial derivative of a function with respect to x is the derivative the function treating all other variables than x as constants. Learn how to calculate the partial derivative along with examples and formulas.
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Boilerpipe Text	The partial derivative of a function (in two or more variables) is its derivative with respect to one of the variables keeping all the other variables as constants. The process of calculating partial derivative is as same as that of an ordinary derivative except we consider the other variables than the variable with respect to which we are differentiating as constants. Let us learn more about how to calculate partial derivatives of different orders along with examples. 1. What is Partial Derivative? 2. Partial Derivative Symbol 3. How to Calculate Partial Derivatives? 4. Partial Derivatives of Different Orders 5. Partial Differentiation Formulas 6. FAQs on Partial Derivative The partial derivative of a multivariable function, say z = f(x, y), is its derivative with respect to one of the variables, x or y in this case, where the other variables are treated as constants. For example, for finding the partial derivative of f(x, y) with respect to x (which is represented by ∂f / ∂x), y is treated as constant and for finding the partial derivative of f(x, y) with respect to y (which is represented by ∂f / ∂y), x is treated as constant Note that we are not considering all the variables as variables while doing partial differentiation (instead, we are considering only one variable as a variable at a time) and hence the name "partial". The limit definition of a partial derivative looks very similar to the limit definition of the derivative. We can find the partial derivatives using the following limit formulas: ∂f / ∂x = lim h → 0 [ f(x + h, y) - f(x, y) ] / h ∂f / ∂y = lim h → 0 [ f(x, y + h) - f(x, y) ] / h These formulas resemble the derivative definition using the first principle. Example of Partial Derivative If f (x, y) = xy, then find the partial derivative ∂f / ∂x. Solution: ∂f / ∂x = lim h → 0 [ f(x + h, y) - f(x, y) ] / h = lim h → 0 [ (x + h)y - xy ] / h = lim h → 0 [xy + hy - xy] / h = lim h → 0 [hy]/h = lim h → 0 y = y Therefore, ∂f / ∂x = y. We know that the ordinary derivative of a function y = f(x) is denoted by one of the notations dy/dx, d/dx (y), d/dx (f(x)), f '(x), etc. For representing a partial derivative we use the symbol "∂" instead of "d". We pronounce "∂" to be "doh" but it has some other names like "partial", "del", "partial dee", "dee", "Jacobi's delta", etc. If z = f(x, y) is a function in two variables then ∂f / ∂x is the partial derivative of f with respect to x ∂f / ∂y is the partial derivative of f with respect to y Just like how we have different symbols of ordinary derivatives, we have different notations for partial derivatives as well. For example, ∂f / ∂x can be written as f x , f x ', D x f, ∂ / ∂x (f), ∂ x f, ∂ / ∂x [f(x, y)], ∂z / ∂x, etc. We have already seen that the limit definitions are used to find the partial derivatives. But using the limit formula and computing the limit is not always easy. Thus, we have another method to calculate partial derivatives that follow right from its definition. In this method, if z = f(x, y) is the function, then we can compute the partial derivatives using the following steps: Step 1 : Identify the variable with respect to which we have to find the partial derivative. Step 2: Except for the variable found in Step 1 , treat all the other variables as constants. Step 3: Differentiate the function just using the rules of ordinary differentiation. Wait! Read Step 3 again. Yes, the rules of ordinary differentiation are as same as that of partial differentiation. In partial differentiation, just treating variables is different, that's it! Example: Let us solve the same above example (If f (x, y) = xy, then find the partial derivative ∂f / ∂x) using the above steps. Solution: We have to find ∂f / ∂x. It means, we have to find the partial derivative of f with respect to x. So we treat y as constant. Thus, we can write 'y' outside the derivative (as in ordinary differentiation, we have a rule that says d/dx (c y) = c dy/dx, where 'c' is a constant). Thus, ∂f / ∂x = ∂ / ∂x (xy) = y ∂ / ∂x (x) = y (1) (Using power rule , d/dx (x) = 1) = y We have got the same answer as we got using the limit definition. We have derivatives like first-order derivatives (like dy/dx), second-order derivatives (like d 2 y/dx 2 ), etc in ordinary derivatives. Likewise, we have first-order, second-order, and higher-order derivatives in partial derivatives also. First Order Partial Derivatives If z = f(x, y) is a function in two variables, then it can have two first-order partial derivatives, namely ∂f / ∂x and ∂f / ∂y. Example: If z = x 2 + y 2 , find all the first order partial derivatives. Solution: f x = ∂f / ∂x = ∂ / ∂x (x 2 + y 2 ) = ∂ / ∂x (x 2 ) + ∂ / ∂x (y 2 ) = 2x + 0 (as y is a constant) = 2x f y = ∂f / ∂y = ∂ / ∂y (x 2 + y 2 ) = ∂ / ∂y (x 2 ) + ∂ / ∂y (y 2 ) = 0 + 2y (as x is a constant) = 2y Second Order Partial Derivatives The second-order partial derivative is obtained by differentiating the function with respect to the indicated variables successively one after the other. If z = f(x, y) is a function in two variables, then it can have four second-order partial derivatives, namely ∂ 2 f / ∂x 2 , ∂ 2 f / ∂y 2 , ∂ 2 f / ∂x ∂y and ∂ 2 f / ∂y ∂x. To find them, we can first differentiate the function partially with the latter variable, and then partially differentiate the result with respect to the former variable. i.e., f xx = ∂ 2 f / ∂x 2 = ∂ / ∂x (∂f / ∂x) = ∂ / ∂x (f x ) f yy = ∂ 2 f / ∂y 2 = ∂ / ∂y (∂f / ∂y) = ∂ / ∂x (f y ) f yx = ∂ 2 f / ∂x ∂y = ∂ / ∂x (∂f / ∂y) = ∂ / ∂x (f y ) f xy = ∂ 2 f / ∂y ∂x = ∂ / ∂y (∂f / ∂x) = ∂ / ∂y (f x ) Observe the notations f yx and f xy . The order of variables in each subscript indicate the order of partial differentiation. For example, f yx means to partially differentiate with respect to y first and then with respect to x, and this is same as ∂ 2 f / ∂x ∂y. Example: If z = x 2 + y 2 , find all the second order partial derivatives. Solution: In the above example, we have already found that f x = 2x and f y = 2y. Now, f xx = ∂ / ∂x (f x ) = ∂ / ∂x (2x) = 2 f yy = ∂ / ∂y (f y ) = ∂ / ∂y (2y) = 2 f yx = ∂ / ∂x (f y ) = ∂ / ∂x (2y) = 0 f xy = ∂ / ∂y (f x ) = ∂ / ∂y (2x) = 0 Now that f yx = f xy . Thus, the order of partial differentiation doesn't matter. The process of finding partial derivatives is known as Partial Differentiation. To find the first-order partial derivatives (as discussed earlier) of a function z = f(x, y) we use the following limit formulas: ∂f / ∂x = lim h → 0 [ f(x + h, y) - f(x, y) ] / h ∂f / ∂y = lim h → 0 [ f(x, y + h) - f(x, y) ] / h But instead of using these formulas, just treating all the other variables than the variable with respect to which we are partially differentiating as constants would make the process of partial differentiation very easier. In this process, we just use the same rules as ordinary differentiation and among them, the important rules are as follows: Power Rule The power rule of differentiation says d/dx (x n ) = n x n-1 . The same rule can be applied in partial derivatives also. Example: ∂ / ∂x (x 2 y) = y ∂ / ∂x (x 2 ) = y (2x) = 2xy. Product Rule The product rule of ordinary differentiation says d/dx (uv) = u dv/dx + v du/dx. We can apply the same rule in partial differentiation as well when there are two functions of the same variable. Example: ∂ / ∂x (xy sin x) = y ∂ / ∂x (x sin x) = y [ x ∂ / ∂x (sin x) + sin x ∂ / ∂x (x) ] = y [ x cos x + sin x] Quotient Rule The quotient rule of ordinary differentiation says d/dx (u/v) = [ v du/dx - u dv/dx ] / v 2 . As other rules, this rule can be applied for finding partial derivatives also. Example: ∂ / ∂x ( xy / sin x) = y ∂ / ∂x (x / sin x) = y [ ( sin x ∂ / ∂x (x) - x ∂ / ∂x (sin x) ) / sin 2 x ] = y [ sin x - x cos x] / sin 2 x Chain Rule of Partial Differentiation The chain rule is used when we have to differentiate an implicit function . The chain rule of partial derivatives works a little differently when compared to ordinary derivatives. Sometimes, the rule involves both partial derivatives and ordinary derivatives. There are various forms of this rule and the application of one of them depends upon how each variable of the function is defined. If y = f(x) is a function where x is again a function of two variables u and v (i.e., x = x (u, v)) then ∂f/∂u = ∂f/∂x · ∂x/∂u; ∂f/∂v = ∂f/∂x · ∂x/∂v If z = f(x, y), where each of x and y are again functions of a variable t (i.e., x = x(t) and y = y(t)) then df/dt = (∂f/∂x · dx/dt) + (∂f/∂y · dy/dt) If z = f(x, y) is a function and each of x and y are again functions of two variables u and v (i.e., x = x(u, v) and y = y(u, v)) then ∂f/∂u = ∂f/∂x · ∂x/∂u + ∂f/∂y · ∂y/∂u; ∂f/∂v = ∂f/∂x · ∂x/∂v + ∂f/∂y · ∂y/∂v Example: If z = e xy , where x = uv and y = u + v then find the partial derivative ∂f/∂u. Solution: By the chain rule of partial derivatives: ∂f/∂u = ∂f/∂x · ∂x/∂u + ∂f/∂y · ∂y/∂u = ∂ / ∂x (e xy ) · ∂ / ∂u (uv) + ∂ / ∂y (e xy ) · ∂ / ∂u (u + v) = (e xy · y) (v) + (e xy · x) (1) = e xy (x + vy) Other Rules of Partial Differentiation If f(x, y) = a constant, then the following formula gives the relation between the ordinary derivative and the partial derivatives which follows from implicit differentiation . dy/dx = -f x /f y . For any two functions u(x, y) and v(x, y), the determinant \| ∂ u / ∂ x ∂ u / ∂ y ∂ v / ∂ x ∂ v / ∂ y \| is known as Jacobian of u and v. The Laplace equation of partial derivatives is ∂ 2 f / ∂x 2 + ∂ 2 f / ∂y 2 + ∂ 2 f / ∂z 2 = 0 where f(x, y, z) is a function in three variables. Any function f that satisfies the Laplace equation is known as the harmonic function. Important Notes on Partial Derivatives: While finding the partial derivative with respect to a variable, all the other variables should be considered as constants. The order of taking derivatives doesn't matter in partial derivatives. i.e., ∂ 2 f / ∂x ∂y = ∂ 2 f / ∂y ∂x. The rules of derivatives apply for partial differentiation as well. Instead of using the limit definition, applying derivative formulas make the process of finding the partial derivatives easier. ☛ Related Topics: Partial Differential Equations Partial Derivative Calculator Read More
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![Download the Cuemath App](https://static.qumath.in/static/next/_next/static/images/app_banner_desktop-8753ba6fb9ffbf8a558f26e1450c864e.webp) ![Download the Cuemath App](https://static.qumath.in/static/next/_next/static/images/app_banner_mobile-3f26e4236459d2a093f12eabcd31dc32.webp) ![Download the Cuemath App](https://static.qumath.in/static/next/_next/static/images/app_banner_desktop-8753ba6fb9ffbf8a558f26e1450c864e.webp) ![Download the Cuemath App](https://static.qumath.in/static/next/_next/static/images/app_banner_mobile-3f26e4236459d2a093f12eabcd31dc32.webp) [Learn]() [Practice]()[Download]() # Partial Derivative The partial derivative of a function (in two or more variables) is its derivative with respect to one of the variables keeping all the other variables as constants. The process of calculating partial derivative is as same as that of an ordinary derivative except we consider the other variables than the variable with respect to which we are differentiating as constants. Let us learn more about how to calculate partial derivatives of different orders along with examples. \| \| \| \|---\|---\| \| 1\. \| [What is Partial Derivative?](https://www.cuemath.com/calculus/partial-derivative/#a) \| \| 2\. \| [Partial Derivative Symbol](https://www.cuemath.com/calculus/partial-derivative/#b) \| \| 3\. \| [How to Calculate Partial Derivatives?](https://www.cuemath.com/calculus/partial-derivative/#c) \| \| 4\. \| [Partial Derivatives of Different Orders](https://www.cuemath.com/calculus/partial-derivative/#d) \| \| 5\. \| [Partial Differentiation Formulas](https://www.cuemath.com/calculus/partial-derivative/#e) \| \| 6\. \| [FAQs on Partial Derivative](https://www.cuemath.com/calculus/partial-derivative/#faqs) \| ## What is Partial Derivative? The partial derivative of a multivariable function, say z = f(x, y), is its [derivative](https://www.cuemath.com/calculus/differentiable/) with respect to one of the variables, x or y in this case, where the other variables are treated as constants. For example, - for finding the partial derivative of f(x, y) with respect to x (which is represented by ∂f / ∂x), y is treated as constant and - for finding the partial derivative of f(x, y) with respect to y (which is represented by ∂f / ∂y), x is treated as constant Note that we are not considering all the variables as variables while doing partial differentiation (instead, we are considering only one variable as a variable at a time) and hence the name "partial". The limit definition of a partial derivative looks very similar to the [limit](https://www.cuemath.com/calculus/limits/) definition of the derivative. We can find the partial derivatives using the following limit formulas: - ∂f / ∂x = lim h → 0 \[ f(x + h, y) - f(x, y) \] / h - ∂f / ∂y = lim h → 0 \[ f(x, y + h) - f(x, y) \] / h ![partial derivative formula](https://d138zd1ktt9iqe.cloudfront.net/media/seo_landing_files/partial-derivative-formula-1648621191.png) These formulas resemble the derivative definition using the first principle. ### Example of Partial Derivative If f (x, y) = xy, then find the partial derivative ∂f / ∂x. Solution: ∂f / ∂x = lim h → 0 \[ f(x + h, y) - f(x, y) \] / h = lim h → 0 \[ (x + h)y - xy \] / h = lim h → 0 \[xy + hy - xy\] / h = lim h → 0 \[hy\]/h = lim h → 0 y = y Therefore, ∂f / ∂x = y. ## Partial Derivative Symbol We know that the ordinary derivative of a function y = f(x) is denoted by one of the notations dy/dx, d/dx (y), d/dx (f(x)), f '(x), etc. For representing a partial derivative we use the symbol "∂" instead of "d". We pronounce "∂" to be "doh" but it has some other names like "partial", "del", "partial dee", "dee", "Jacobi's delta", etc. If z = f(x, y) is a function in two variables then - ∂f / ∂x is the partial derivative of f with respect to x - ∂f / ∂y is the partial derivative of f with respect to y Just like how we have different [symbols](https://www.cuemath.com/numbers/math-symbols/) of ordinary derivatives, we have different notations for partial derivatives as well. For example, ∂f / ∂x can be written as fx, fx', Dxf, ∂ / ∂x (f), ∂x f, ∂ / ∂x \[f(x, y)\], ∂z / ∂x, etc. ## Calculate Partial Derivatives We have already seen that the limit definitions are used to find the partial derivatives. But using the [limit formula](https://www.cuemath.com/limit-formula/) and computing the limit is not always easy. Thus, we have another method to calculate partial derivatives that follow right from its definition. In this method, if z = f(x, y) is the function, then we can compute the partial derivatives using the following steps: - Step 1: Identify the variable with respect to which we have to find the partial derivative. - Step 2: Except for the variable found in Step 1, treat all the other variables as constants. - Step 3: Differentiate the function just using the rules of ordinary differentiation. Wait! Read Step 3 again. Yes, the rules of ordinary differentiation are as same as that of partial differentiation. In partial differentiation, just treating variables is different, that's it\! Example: Let us solve the same above example (If f (x, y) = xy, then find the partial derivative ∂f / ∂x) using the above steps. Solution: We have to find ∂f / ∂x. It means, we have to find the partial derivative of f with respect to x. So we treat y as constant. Thus, we can write 'y' outside the derivative (as in ordinary differentiation, we have a rule that says d/dx (c y) = c dy/dx, where 'c' is a constant). Thus, ∂f / ∂x = ∂ / ∂x (xy) = y ∂ / ∂x (x) = y (1) (Using [power rule](https://www.cuemath.com/calculus/power-rule/), d/dx (x) = 1) = y We have got the same answer as we got using the limit definition. ## Partial Derivatives of Different Orders We have derivatives like first-order derivatives (like dy/dx), [second-order derivatives](https://www.cuemath.com/calculators/second-derivative-calculator/) (like d2y/dx2), etc in ordinary derivatives. Likewise, we have first-order, second-order, and higher-order derivatives in partial derivatives also. ### First Order Partial Derivatives If z = f(x, y) is a function in two variables, then it can have two first-order partial derivatives, namely ∂f / ∂x and ∂f / ∂y. Example: If z = x2 + y2, find all the first order partial derivatives. Solution: fx = ∂f / ∂x = ∂ / ∂x (x2 + y2) = ∂ / ∂x (x2) + ∂ / ∂x (y2) = 2x + 0 (as y is a constant) = 2x fy = ∂f / ∂y = ∂ / ∂y (x2 + y2) = ∂ / ∂y (x2) + ∂ / ∂y (y2) = 0 + 2y (as x is a constant) = 2y ### Second Order Partial Derivatives The second-order partial derivative is obtained by differentiating the function with respect to the indicated variables successively one after the other. If z = f(x, y) is a function in two variables, then it can have four second-order partial derivatives, namely ∂2f / ∂x2, ∂2f / ∂y2, ∂2f / ∂x ∂y and ∂2f / ∂y ∂x. To find them, we can first differentiate the function partially with the latter variable, and then partially differentiate the result with respect to the former variable. i.e., - fxx = ∂2f / ∂x2 = ∂ / ∂x (∂f / ∂x) = ∂ / ∂x (fx) - fyy = ∂2f / ∂y2 = ∂ / ∂y (∂f / ∂y) = ∂ / ∂x (fy) - fyx = ∂2f / ∂x ∂y = ∂ / ∂x (∂f / ∂y) = ∂ / ∂x (fy) - fxy = ∂2f / ∂y ∂x = ∂ / ∂y (∂f / ∂x) = ∂ / ∂y (fx) Observe the notations fyx and fxy. The order of variables in each subscript indicate the order of partial differentiation. For example, fyx means to partially differentiate with respect to y first and then with respect to x, and this is same as ∂2f / ∂x ∂y. Example: If z = x2 + y2, find all the second order partial derivatives. Solution: In the above example, we have already found that fx = 2x and fy = 2y. Now, fxx = ∂ / ∂x (fx) = ∂ / ∂x (2x) = 2 fyy = ∂ / ∂y (fy) = ∂ / ∂y (2y) = 2 fyx = ∂ / ∂x (fy) = ∂ / ∂x (2y) = 0 fxy = ∂ / ∂y (fx) = ∂ / ∂y (2x) = 0 Now that fyx = fxy. Thus, the order of partial differentiation doesn't matter. ## Partial Differentiation Formulas The process of finding partial derivatives is known as Partial Differentiation. To find the first-order partial derivatives (as discussed earlier) of a function z = f(x, y) we use the following limit formulas: - ∂f / ∂x = lim h → 0 \[ f(x + h, y) - f(x, y) \] / h - ∂f / ∂y = lim h → 0 \[ f(x, y + h) - f(x, y) \] / h But instead of using these formulas, just treating all the other variables than the variable with respect to which we are partially differentiating as constants would make the process of partial differentiation very easier. In this process, we just use the same [rules as ordinary differentiation](https://www.cuemath.com/calculus/derivative-rules/) and among them, the important rules are as follows: ### Power Rule The power rule of differentiation says d/dx (xn) = n xn-1. The same rule can be applied in partial derivatives also. Example: ∂ / ∂x (x2y) = y ∂ / ∂x (x2) = y (2x) = 2xy. ### Product Rule The [product rule](https://www.cuemath.com/calculus/product-rule/) of ordinary differentiation says d/dx (uv) = u dv/dx + v du/dx. We can apply the same rule in partial differentiation as well when there are two functions of the same variable. Example: ∂ / ∂x (xy sin x) = y ∂ / ∂x (x sin x) = y \[ x ∂ / ∂x (sin x) + sin x ∂ / ∂x (x) \] = y \[ x cos x + sin x\] ### Quotient Rule The [quotient rule](https://www.cuemath.com/calculus/quotient-rule/) of ordinary differentiation says d/dx (u/v) = \[ v du/dx - u dv/dx \] / v2. As other rules, this rule can be applied for finding partial derivatives also. Example: ∂ / ∂x ( xy / sin x) = y ∂ / ∂x (x / sin x) = y \[ ( sin x ∂ / ∂x (x) - x ∂ / ∂x (sin x) ) / sin2x \] = y \[ sin x - x cos x\] / sin2x ### Chain Rule of Partial Differentiation The chain rule is used when we have to differentiate an [implicit function](https://www.cuemath.com/algebra/implicit-function/). The chain rule of partial derivatives works a little differently when compared to ordinary derivatives. Sometimes, the rule involves both partial derivatives and ordinary derivatives. There are various forms of this rule and the application of one of them depends upon how each variable of the function is defined. - If y = f(x) is a function where x is again a function of two variables u and v (i.e., x = x (u, v)) then ∂f/∂u = ∂f/∂x · ∂x/∂u; ∂f/∂v = ∂f/∂x · ∂x/∂v - If z = f(x, y), where each of x and y are again functions of a variable t (i.e., x = x(t) and y = y(t)) then df/dt = (∂f/∂x · dx/dt) + (∂f/∂y · dy/dt) - If z = f(x, y) is a function and each of x and y are again functions of two variables u and v (i.e., x = x(u, v) and y = y(u, v)) then ∂f/∂u = ∂f/∂x · ∂x/∂u + ∂f/∂y · ∂y/∂u; ∂f/∂v = ∂f/∂x · ∂x/∂v + ∂f/∂y · ∂y/∂v Example: If z = exy, where x = uv and y = u + v then find the partial derivative ∂f/∂u. Solution: By the chain rule of partial derivatives: ∂f/∂u = ∂f/∂x · ∂x/∂u + ∂f/∂y · ∂y/∂u = ∂ / ∂x (exy) · ∂ / ∂u (uv) + ∂ / ∂y (exy) · ∂ / ∂u (u + v) = (exy · y) (v) + (exy · x) (1) = exy (x + vy) ### Other Rules of Partial Differentiation - If f(x, y) = a constant, then the following formula gives the relation between the ordinary derivative and the partial derivatives which follows from [implicit differentiation](https://www.cuemath.com/calculus/implicit-differentiation/). dy/dx = -fx/fy. - For any two functions u(x, y) and v(x, y), the [determinant](https://www.cuemath.com/algebra/determinant-of-matrix/) ∣∣∣∂u/∂x∂u/∂y∂v/∂x∂v/∂y∣∣∣ \\| ∂ u / ∂ x ∂ u / ∂ y ∂ v / ∂ x ∂ v / ∂ y \\| is known as Jacobian of u and v. - The Laplace equation of partial derivatives is ∂2f / ∂x2 + ∂2f / ∂y2 + ∂2f / ∂z2 = 0 where f(x, y, z) is a function in three variables. Any function f that satisfies the Laplace equation is known as the harmonic function. Important Notes on Partial Derivatives: - While finding the partial derivative with respect to a variable, all the other variables should be considered as constants. - The order of taking derivatives doesn't matter in partial derivatives. i.e., ∂2f / ∂x ∂y = ∂2f / ∂y ∂x. - The rules of derivatives apply for partial differentiation as well. - Instead of using the limit definition, applying [derivative formulas](https://www.cuemath.com/derivative-formula/) make the process of finding the partial derivatives easier. ☛ Related Topics: - [Partial Differential Equations](https://www.cuemath.com/calculus/partial-differential-equations/) - [Partial Derivative Calculator](https://www.cuemath.com/calculators/partial-derivative-calculator/) Read More Explore math program Download FREE Study Materials ![](data:image/svg+xml;base64,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) SHEETS ![](https://d138zd1ktt9iqe.cloudfront.net/media/seo_landing_files/thumbnails/partial-derivative-worksheet-newui-first-page-1649086216.jpg) Partial Derivative Worksheet Calculus Worksheet ![](data:image/svg+xml;base64,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) ## Partial Derivative Examples 1. Example 1: Find all the first order partial derivatives of the function f(x, y) = ax2 + 2hxy + by2. Solution: The first-order partial derivatives are: fx = ∂f / ∂x = ∂ / ∂x (ax2 + 2hxy + by2) = ∂ / ∂x (ax2) + ∂ / ∂x (2hxy) + ∂ / ∂x (by2) = 2ax + 2hy (1) + 0 (as y is a constant) = 2ax + 2hy fx = ∂f / ∂x = ∂ / ∂x (ax2 + 2hxy + by2) = ∂ / ∂y (ax2) + ∂ / ∂y (2hxy) + ∂ / ∂y (by2) = 0 + 2hx (1) + 2by (as x is a constant) = 2hx + 2by Answer: fx = 2ax + 2hy and fy = 2hx + 2by. 2. Example 2: Find all the second-order partial derivatives of the function given in Example 1. Solution: We found in Example 1 thatfx = 2ax + 2hy and fy = 2hx + 2by. Now, we will find the second-order partial derivatives. fxx = ∂ / ∂x (fx) = ∂ / ∂x (2ax + 2hy) = 2a fyy = ∂ / ∂y (fy) = ∂ / ∂y (2hx + 2by) = 2b fyx = ∂ / ∂x (fy) = ∂ / ∂x (2hx + 2by) = 2h fxy = ∂ / ∂y (fx) = ∂ / ∂y (2ax + 2hy) = 2h Answer: We found that fxx = 2a, fyy = 2b, fyx = 2h, and fxy = 2h. 3. Example 3: Does the order of differentiation matter while finding the partial derivatives? Justify your answer using Example 2. Solution: From Example 2, fyx = 2h and fxy = 2h. Thus, fxy = fxy. Thus, the order of partial differentiation doesn't matter. Answer: The order doesn't matter as fyx = fxy. View Answer \> ![](https://d138zd1ktt9iqe.cloudfront.net/media/seo_landing_files/caucasian-kid-and-indian-teacher-grade-4-7-1619081163.png) Great learning in high school using simple cues Indulging in rote learning, you are likely to forget concepts. With Cuemath, you will learn visually and be surprised by the outcomes. [Book a Free Trial Class](https://www.cuemath.com/parent/signup/) ## Practice Questions on Partial Differentiation Check Answer \> ## FAQs on Partial Derivative ### How to Calculate Partial Derivatives? We use partial derivatives when the function has more than one variable. If a function f is in terms of two variables x and y, then we can calculate the partial derivatives as follows. - the partial derivative of f = ∂f/∂x and y has to be treated as constant here. - the partial derivative of f = ∂f/∂y and x has to be treated as constant here. ### What is the Symbol of Partial Derivatives? We use the symbol ∂ to represent a partial derivative. For example, the partial derivative of a function f(x, y) with respect to x is written as ∂f/∂x. ### What is the Difference Between Differentiation and Partial Differentiation? We talk about the [derivative](https://www.cuemath.com/calculus/derivatives/) of a function if a function has only one variable in it. For example, the derivative of a function y = f(x) is denoted by df/dx. We talk about partial derivatives when a function z = f(x, y) has more than one variable. The partial derivative of f with respect to x is denoted by ∂f/∂x and while finding this, we treat y as constant. ### What is the Formula Used to Find the Partial Derivative? The partial derivatives of a function z = f(x, y) can be found using the limit formulas: - ∂f / ∂x = lim h → 0 \[ f(x + h, y) - f(x, y) \] / h - ∂f / ∂y = lim h → 0 \[ f(x, y + h) - f(x, y) \] / h ### What Does Partial Derivative Tell Us? Partial derivative tells us to [differentiate](https://www.cuemath.com/calculus/differentiation/) a function partially. It means if we are differentiating partially with respect to one variable, then the remaining [variables](https://www.cuemath.com/algebra/variables-constants-and-expressions/) of the function must be treated as constants. ### What is the Chain Rule of Partial Derivatives? The chain rule of partial derivative is mentioned below: If z = f(x, y) is a [function](https://www.cuemath.com/calculus/What-are-functions/) where x and y are functions of two variables u and v (i.e., x = x(u, v) and y = y(u, v)) then by the chain rule of partial derivatives, - ∂f/∂u = ∂f/∂x · ∂x/∂u + ∂f/∂y · ∂y/∂u - ∂f/∂v = ∂f/∂x · ∂x/∂v + ∂f/∂y · ∂y/∂v ### What is the Formula that Connects Normal Derivatives with Partial Derivatives? When f(x, y) = c, where 'c' is a [constant](https://www.cuemath.com/algebra/constants/), then dy/dx = -fx/fy, where - fx is the partial derivative of f with respect to x - fy is the partial derivative of f with respect to y Explore math program Math worksheets and visual curriculum Download App Become MathFit™: Boost math skills with daily fun challenges and puzzles. Download the app STRATEGY GAMES LOGIC PUZZLES MENTAL MATH Become MathFit™: Boost math skills with daily fun challenges and puzzles. 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Readable Markdown	The partial derivative of a function (in two or more variables) is its derivative with respect to one of the variables keeping all the other variables as constants. The process of calculating partial derivative is as same as that of an ordinary derivative except we consider the other variables than the variable with respect to which we are differentiating as constants. Let us learn more about how to calculate partial derivatives of different orders along with examples. \| \| \| \|---\|---\| \| 1\. \| [What is Partial Derivative?](https://www.cuemath.com/calculus/partial-derivative/#a) \| \| 2\. \| [Partial Derivative Symbol](https://www.cuemath.com/calculus/partial-derivative/#b) \| \| 3\. \| [How to Calculate Partial Derivatives?](https://www.cuemath.com/calculus/partial-derivative/#c) \| \| 4\. \| [Partial Derivatives of Different Orders](https://www.cuemath.com/calculus/partial-derivative/#d) \| \| 5\. \| [Partial Differentiation Formulas](https://www.cuemath.com/calculus/partial-derivative/#e) \| \| 6\. \| [FAQs on Partial Derivative](https://www.cuemath.com/calculus/partial-derivative/#faqs) \| The partial derivative of a multivariable function, say z = f(x, y), is its [derivative](https://www.cuemath.com/calculus/differentiable/) with respect to one of the variables, x or y in this case, where the other variables are treated as constants. For example, - for finding the partial derivative of f(x, y) with respect to x (which is represented by ∂f / ∂x), y is treated as constant and - for finding the partial derivative of f(x, y) with respect to y (which is represented by ∂f / ∂y), x is treated as constant Note that we are not considering all the variables as variables while doing partial differentiation (instead, we are considering only one variable as a variable at a time) and hence the name "partial". The limit definition of a partial derivative looks very similar to the [limit](https://www.cuemath.com/calculus/limits/) definition of the derivative. We can find the partial derivatives using the following limit formulas: - ∂f / ∂x = lim h → 0 \[ f(x + h, y) - f(x, y) \] / h - ∂f / ∂y = lim h → 0 \[ f(x, y + h) - f(x, y) \] / h These formulas resemble the derivative definition using the first principle. ### Example of Partial Derivative If f (x, y) = xy, then find the partial derivative ∂f / ∂x. Solution: ∂f / ∂x = lim h → 0 \[ f(x + h, y) - f(x, y) \] / h = lim h → 0 \[ (x + h)y - xy \] / h = lim h → 0 \[xy + hy - xy\] / h = lim h → 0 \[hy\]/h = lim h → 0 y = y Therefore, ∂f / ∂x = y. We know that the ordinary derivative of a function y = f(x) is denoted by one of the notations dy/dx, d/dx (y), d/dx (f(x)), f '(x), etc. For representing a partial derivative we use the symbol "∂" instead of "d". We pronounce "∂" to be "doh" but it has some other names like "partial", "del", "partial dee", "dee", "Jacobi's delta", etc. If z = f(x, y) is a function in two variables then - ∂f / ∂x is the partial derivative of f with respect to x - ∂f / ∂y is the partial derivative of f with respect to y Just like how we have different [symbols](https://www.cuemath.com/numbers/math-symbols/) of ordinary derivatives, we have different notations for partial derivatives as well. For example, ∂f / ∂x can be written as fx, fx', Dxf, ∂ / ∂x (f), ∂x f, ∂ / ∂x \[f(x, y)\], ∂z / ∂x, etc. We have already seen that the limit definitions are used to find the partial derivatives. But using the [limit formula](https://www.cuemath.com/limit-formula/) and computing the limit is not always easy. Thus, we have another method to calculate partial derivatives that follow right from its definition. In this method, if z = f(x, y) is the function, then we can compute the partial derivatives using the following steps: - Step 1: Identify the variable with respect to which we have to find the partial derivative. - Step 2: Except for the variable found in Step 1, treat all the other variables as constants. - Step 3: Differentiate the function just using the rules of ordinary differentiation. Wait! Read Step 3 again. Yes, the rules of ordinary differentiation are as same as that of partial differentiation. In partial differentiation, just treating variables is different, that's it\! Example: Let us solve the same above example (If f (x, y) = xy, then find the partial derivative ∂f / ∂x) using the above steps. Solution: We have to find ∂f / ∂x. It means, we have to find the partial derivative of f with respect to x. So we treat y as constant. Thus, we can write 'y' outside the derivative (as in ordinary differentiation, we have a rule that says d/dx (c y) = c dy/dx, where 'c' is a constant). Thus, ∂f / ∂x = ∂ / ∂x (xy) = y ∂ / ∂x (x) = y (1) (Using [power rule](https://www.cuemath.com/calculus/power-rule/), d/dx (x) = 1) = y We have got the same answer as we got using the limit definition. We have derivatives like first-order derivatives (like dy/dx), [second-order derivatives](https://www.cuemath.com/calculators/second-derivative-calculator/) (like d2y/dx2), etc in ordinary derivatives. Likewise, we have first-order, second-order, and higher-order derivatives in partial derivatives also. ### First Order Partial Derivatives If z = f(x, y) is a function in two variables, then it can have two first-order partial derivatives, namely ∂f / ∂x and ∂f / ∂y. Example: If z = x2 + y2, find all the first order partial derivatives. Solution: fx = ∂f / ∂x = ∂ / ∂x (x2 + y2) = ∂ / ∂x (x2) + ∂ / ∂x (y2) = 2x + 0 (as y is a constant) = 2x fy = ∂f / ∂y = ∂ / ∂y (x2 + y2) = ∂ / ∂y (x2) + ∂ / ∂y (y2) = 0 + 2y (as x is a constant) = 2y ### Second Order Partial Derivatives The second-order partial derivative is obtained by differentiating the function with respect to the indicated variables successively one after the other. If z = f(x, y) is a function in two variables, then it can have four second-order partial derivatives, namely ∂2f / ∂x2, ∂2f / ∂y2, ∂2f / ∂x ∂y and ∂2f / ∂y ∂x. To find them, we can first differentiate the function partially with the latter variable, and then partially differentiate the result with respect to the former variable. i.e., - fxx = ∂2f / ∂x2 = ∂ / ∂x (∂f / ∂x) = ∂ / ∂x (fx) - fyy = ∂2f / ∂y2 = ∂ / ∂y (∂f / ∂y) = ∂ / ∂x (fy) - fyx = ∂2f / ∂x ∂y = ∂ / ∂x (∂f / ∂y) = ∂ / ∂x (fy) - fxy = ∂2f / ∂y ∂x = ∂ / ∂y (∂f / ∂x) = ∂ / ∂y (fx) Observe the notations fyx and fxy. The order of variables in each subscript indicate the order of partial differentiation. For example, fyx means to partially differentiate with respect to y first and then with respect to x, and this is same as ∂2f / ∂x ∂y. Example: If z = x2 + y2, find all the second order partial derivatives. Solution: In the above example, we have already found that fx = 2x and fy = 2y. Now, fxx = ∂ / ∂x (fx) = ∂ / ∂x (2x) = 2 fyy = ∂ / ∂y (fy) = ∂ / ∂y (2y) = 2 fyx = ∂ / ∂x (fy) = ∂ / ∂x (2y) = 0 fxy = ∂ / ∂y (fx) = ∂ / ∂y (2x) = 0 Now that fyx = fxy. Thus, the order of partial differentiation doesn't matter. The process of finding partial derivatives is known as Partial Differentiation. To find the first-order partial derivatives (as discussed earlier) of a function z = f(x, y) we use the following limit formulas: - ∂f / ∂x = lim h → 0 \[ f(x + h, y) - f(x, y) \] / h - ∂f / ∂y = lim h → 0 \[ f(x, y + h) - f(x, y) \] / h But instead of using these formulas, just treating all the other variables than the variable with respect to which we are partially differentiating as constants would make the process of partial differentiation very easier. In this process, we just use the same [rules as ordinary differentiation](https://www.cuemath.com/calculus/derivative-rules/) and among them, the important rules are as follows: ### Power Rule The power rule of differentiation says d/dx (xn) = n xn-1. The same rule can be applied in partial derivatives also. Example: ∂ / ∂x (x2y) = y ∂ / ∂x (x2) = y (2x) = 2xy. ### Product Rule The [product rule](https://www.cuemath.com/calculus/product-rule/) of ordinary differentiation says d/dx (uv) = u dv/dx + v du/dx. We can apply the same rule in partial differentiation as well when there are two functions of the same variable. Example: ∂ / ∂x (xy sin x) = y ∂ / ∂x (x sin x) = y \[ x ∂ / ∂x (sin x) + sin x ∂ / ∂x (x) \] = y \[ x cos x + sin x\] ### Quotient Rule The [quotient rule](https://www.cuemath.com/calculus/quotient-rule/) of ordinary differentiation says d/dx (u/v) = \[ v du/dx - u dv/dx \] / v2. As other rules, this rule can be applied for finding partial derivatives also. Example: ∂ / ∂x ( xy / sin x) = y ∂ / ∂x (x / sin x) = y \[ ( sin x ∂ / ∂x (x) - x ∂ / ∂x (sin x) ) / sin2x \] = y \[ sin x - x cos x\] / sin2x ### Chain Rule of Partial Differentiation The chain rule is used when we have to differentiate an [implicit function](https://www.cuemath.com/algebra/implicit-function/). The chain rule of partial derivatives works a little differently when compared to ordinary derivatives. Sometimes, the rule involves both partial derivatives and ordinary derivatives. There are various forms of this rule and the application of one of them depends upon how each variable of the function is defined. - If y = f(x) is a function where x is again a function of two variables u and v (i.e., x = x (u, v)) then ∂f/∂u = ∂f/∂x · ∂x/∂u; ∂f/∂v = ∂f/∂x · ∂x/∂v - If z = f(x, y), where each of x and y are again functions of a variable t (i.e., x = x(t) and y = y(t)) then df/dt = (∂f/∂x · dx/dt) + (∂f/∂y · dy/dt) - If z = f(x, y) is a function and each of x and y are again functions of two variables u and v (i.e., x = x(u, v) and y = y(u, v)) then ∂f/∂u = ∂f/∂x · ∂x/∂u + ∂f/∂y · ∂y/∂u; ∂f/∂v = ∂f/∂x · ∂x/∂v + ∂f/∂y · ∂y/∂v Example: If z = exy, where x = uv and y = u + v then find the partial derivative ∂f/∂u. Solution: By the chain rule of partial derivatives: ∂f/∂u = ∂f/∂x · ∂x/∂u + ∂f/∂y · ∂y/∂u = ∂ / ∂x (exy) · ∂ / ∂u (uv) + ∂ / ∂y (exy) · ∂ / ∂u (u + v) = (exy · y) (v) + (exy · x) (1) = exy (x + vy) ### Other Rules of Partial Differentiation - If f(x, y) = a constant, then the following formula gives the relation between the ordinary derivative and the partial derivatives which follows from [implicit differentiation](https://www.cuemath.com/calculus/implicit-differentiation/). dy/dx = -fx/fy. - For any two functions u(x, y) and v(x, y), the [determinant](https://www.cuemath.com/algebra/determinant-of-matrix/) \\| ∂ u / ∂ x ∂ u / ∂ y ∂ v / ∂ x ∂ v / ∂ y \\| is known as Jacobian of u and v. - The Laplace equation of partial derivatives is ∂2f / ∂x2 + ∂2f / ∂y2 + ∂2f / ∂z2 = 0 where f(x, y, z) is a function in three variables. Any function f that satisfies the Laplace equation is known as the harmonic function. Important Notes on Partial Derivatives: - While finding the partial derivative with respect to a variable, all the other variables should be considered as constants. - The order of taking derivatives doesn't matter in partial derivatives. i.e., ∂2f / ∂x ∂y = ∂2f / ∂y ∂x. - The rules of derivatives apply for partial differentiation as well. - Instead of using the limit definition, applying [derivative formulas](https://www.cuemath.com/derivative-formula/) make the process of finding the partial derivatives easier. ☛ Related Topics: - [Partial Differential Equations](https://www.cuemath.com/calculus/partial-differential-equations/) - [Partial Derivative Calculator](https://www.cuemath.com/calculators/partial-derivative-calculator/) Read More
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