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Raw Queries and Responses

1. Shard Calculation

Query:

Response:

Calculated Shard: 65 (from laksa044)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa065.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://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM14-3-8.php\')), getAhrefsUnparsedNoserviceFromURL(\'https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM14-3-8.php\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM14-3-8.php","crawl_time":1774421374,"first_indexed_time":1561509881,"http_code":200,"src_unparsed":"edu,utexas!ma,web,\/users\/m408s\/m408d\/CurrentWeb\/LM14-3-8.php s443","src_root_hash":"16797036158535431465","history_drop_reason":null,"meta_title":"Partial Derivatives","meta_descriptions":[],"attrs_boilerpipe_text":"Higher Order Derivatives\nSince\nf\nx\nand\nf\ny\nare functions of\nx\nand\ny\n,\nwe can take derivatives of\nf\nx\nand\nf\ny\nto\n          get second derivatives\n.  There are four such\n      second derivatives, since each time we can differentiate with\n      respect to\nx\nor\ny\n.  Each of these second derivatives has\n      multiple notations, and we have listed some of them.\nNotation for second partial\n                  derivatives\n(\nf\nx\n)\nx\n=\nf\nx\nx\n=\nf\n11\n=\n∂\n∂\nx\n(\n∂\nf\n∂\nx\n)\n=\n∂\n2\nf\n∂\nx\n2\n=\n∂\n2\nz\n∂\nx\n2\n(\nf\nx\n)\ny\n=\nf\nx\ny\n=\nf\n12\n=\n∂\n∂\ny\n(\n∂\nf\n∂\nx\n)\n=\n∂\n2\nf\n∂\ny\n∂\nx\n=\n∂\n2\nz\n∂\ny\n∂\nx\n(\nf\ny\n)\nx\n=\nf\ny\nx\n=\nf\n21\n=\n∂\n∂\nx\n(\n∂\nf\n∂\ny\n)\n=\n∂\n2\nf\n∂\nx\n∂\ny\n=\n∂\n2\nz\n∂\nx\n∂\ny\n(\nf\ny\n)\ny\n=\nf\ny\ny\n=\nf\n22\n=\n∂\n∂\ny\n(\n∂\nf\n∂\ny\n)\n=\n∂\n2\nf\n∂\ny\n2\n=\n∂\n2\nz\n∂\ny\n2\nA nice result regarding second partial derivatives is\nClairaut's Theorem\n, which tells us\n      that the mixed variable partial derivatives are equal.\nClairaut's Theorem\nIf\nf\nx\ny\nand\nf\ny\nx\nare both defined and continuous in\n            a region containing the point\n(\na\n,\nb\n)\n, then\nf\nx\ny\n(\na\n,\nb\n)\n=\nf\ny\nx\n(\na\n,\nb\n)\n.\nA consequence of this theorem is that we don't need to keep\n      track of the order in which we take derivatives. \nExample 1\n:  Let\nf\n(\nx\n,\ny\n)\n=\n3\nx\n2\n−\n4\ny\n3\n−\n7\nx\n2\ny\n3\n. \n      Previously, we determined that\nf\nx\n=\n6\nx\n−\n14\nx\ny\n3\nand\nf\ny\n=\n−\n12\ny\n2\n−\n21\nx\n2\ny\n2\n.  We have four second derivatives, but\n      as Clairaut's Theorem tells us,\nf\nx\ny\n=\nf\ny\nx\n, so we really only\n      need to compute three of them (we do all four to illustrate the\n      theorem).\nSolution 1\n:\n(\nf\nx\n)\nx\n=\n∂\n∂\nx\nf\nx\n=\n∂\n∂\nx\n(\n6\nx\n−\n14\nx\ny\n3\n)\n=\n6\n−\n14\ny\n3\n(\nf\nx\n)\ny\n=\n∂\n∂\ny\nf\nx\n=\n∂\n∂\ny\n(\n6\nx\n−\n14\nx\ny\n3\n)\n=\n0\n−\n42\nx\ny\n2\n=\n−\n42\nx\ny\n2\n(\nf\ny\n)\nx\n=\n∂\n∂\nx\nf\ny\n=\n∂\n∂\nx\n(\n12\ny\n2\n−\n21\nx\n2\ny\n2\n)\n=\n0\n−\n42\nx\ny\n2\n=\n−\n42\nx\ny\n2\n(\nf\ny\n)\ny\n=\n∂\n∂\ny\nf\ny\n=\n∂\n∂\ny\n(\n12\ny\n2\n−\n21\nx\n2\ny\n2\n)\n=\n24\ny\n−\n42\nx\n2\ny\nHigher partial derivatives and Clairaut's theorem are explained\n      in the following video.","attrs_markdown":"|   |   |   |\n|---|---|---|\n| ![](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/logo.jpg)       [Home](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/index.html)     Integration by Parts [Integration by Parts](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-1-2.php)   [Examples](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-1-3.php)   [Integration by Parts with a definite integral](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-1-5.php)   [Going in Circles](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-1-7.php)   [Tricks of the Trade](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-1-8.php)    Integrals of Trig Functions [Antiderivatives of Basic Trigonometric Functions](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-2-2.php)   [Product of Sines and Cosines (mixed even and odd powers or only odd powers)](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-2-3.php)   [Product of Sines and Cosines (only even powers)](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-2-6.php)   [Product of Secants and Tangents](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-2-8.php)   [Other Cases](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-2-12.php)    Trig Substitutions [How Trig Substitution Works](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-3-2.php)   [Summary of trig substitution options](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-3-3.php)   [Examples](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-3-5.php)   [Completing the Square](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-3-9.php)    Partial Fractions [Introduction to Partial Fractions](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-4-2.php)   [Linear Factors](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-4-3.php)   [Irreducible Quadratic Factors](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-4-6.php)   [Improper Rational Functions and Long Division](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-4-8.php)   [Summary](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-4-9.php)    Strategies of Integration [Substitution](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-5-2.php)   [Integration by Parts](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-5-3.php)   [Trig Integrals](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-5-4.php)   [Trig Substitutions](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-5-5.php)   [Partial Fractions](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-5-6.php)    Improper Integrals [Type 1 - Improper Integrals with Infinite Intervals of Integration](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-8-2.php)   [Type 2 - Improper Integrals with Discontinuous Integrands](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-8-4.php)   [Comparison Tests for Convergence](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM7-8-7.php)    Modeling with Differential Equations [Introduction](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM9-1-2.php)   [Separable Equations](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM9-1-3.php)   [A Second Order Problem](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM9-1-4.php)    Euler's Method and Direction Fields [Euler's Method (follow your nose)](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM9-2-2.php)   [Direction Fields](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM9-2-4.php)   [Euler's method revisited](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM9-2-5.php)    Separable Equations [The Simplest Differential Equations](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM9-3-2.php)   [Separable differential equations](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM9-3-3.php)   [Mixing and Dilution](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM9-3-4.php)    Models of Growth [Exponential Growth and Decay](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM9-4-2.php)   [The Zombie Apocalypse (Logistic Growth)](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM9-4-3.php)    Linear Equations [Linear ODEs: Working an Example](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM9-5-2.php)   [The Solution in General](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM9-5-3.php)   [Saving for Retirement](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM9-5-4.php)    Parametrized Curves [Three kinds of functions, three kinds of curves](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM10-1-2.php)   [The Cycloid](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM10-1-3.php)   [Visualizing Parametrized Curves](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM10-1-4.php)   [Tracing Circles and Ellipses](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM10-1-5.php)   [Lissajous Figures](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM10-1-6.php)    Calculus with Parametrized Curves [Video: Slope and Area](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM10-2-2.php)   [Video: Arclength and Surface Area](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM10-2-3.php)   [Summary and Simplifications](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM10-2-4.php)   [Higher Derivatives](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM10-2-5.php)    Polar Coordinates [Definitions of Polar Coordinates](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM10-3-2.php)   [Graphing polar functions](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM10-3-3.php)   [Video: Computing Slopes of Tangent Lines](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM10-3-4.php)    Areas and Lengths of Polar Curves [Area Inside a Polar Curve](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM10-4-2.php)   [Area Between Polar Curves](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM10-4-3.php)   [Arc Length of Polar Curves](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM10-4-4.php)    Conic sections [Slicing a 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[Examples of Infinite Sequences](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-1-3.php)   [Limit Laws for Sequences](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-1-6.php)   [Theorems for and Examples of Computing Limits of Sequences](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-1-7.php)   [Monotonic Covergence](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-1-10.php)    Infinite Series [Introduction](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-2-2.php)   [Geometric Series](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-2-4.php)   [Limit Laws for Series](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-2-5.php)   [Test for Divergence and Other Theorems](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-2-7.php)   [Telescoping Sums](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-2-9.php)    Integral Test [Preview of Coming Attractions](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-3-2.php)   [The Integral Test](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-3-3.php)   [Estimates for the Value of the Series](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-3-5.php)    Comparison Tests [The Basic Comparison Test](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-4-2.php)   [The Limit Comparison Test](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-4-4.php)    Convergence of Series with Negative Terms [Introduction, Alternating Series,and the AS Test](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-5-2.php)   [Absolute Convergence](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-5-4.php)   [Rearrangements](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-5-8.php)    The Ratio and Root Tests [The Ratio Test](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-6-2.php)   [The Root Test](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-6-4.php)   [Examples](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-6-6.php)    Strategies for testing Series [Strategy to Test Series and a Review of Tests](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-7-2.php)   [Examples, Part 1](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-7-3.php)   [Examples, Part 2](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-7-4.php)    Power Series [Radius and Interval of Convergence](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-8-2.php)   [Finding the Interval of Convergence](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-8-3.php)   [Power Series Centered at x=a x = a](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-8-6.php)    Representing Functions as Power Series [Functions as Power Series](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-9-2.php)   [Derivatives and Integrals of Power Series](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-9-4.php)   [Applications and Examples](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-9-5.php)    Taylor and Maclaurin Series [The Formula for Taylor Series](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-10-2.php)   [Taylor Series for Common Functions](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-10-4.php)   [Adding, Multiplying, and Dividing Power Series](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-10-7.php)   [Miscellaneous Useful Facts](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-10-8.php)    Applications of Taylor Polynomials [Taylor Polynomials](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-11-2.php)   [When Functions Are Equal to Their Taylor Series](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-11-5.php)   [When a Function Does Not Equal Its Taylor Series](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-11-7.php)   [Other Uses of Taylor Polynomials](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM11-11-8.php)    Functions of 2 and 3 variables [Functions of several variables](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM14-1-2.php)   [Limits and continuity](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM14-1-4.php)    Partial Derivatives [One variable at a time (yet again)](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM14-3-2.php)   [Definitions and Examples](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM14-3-3.php)   [An Example from DNA](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM14-3-5.php)   [Geometry of partial derivatives](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM14-3-6.php)   [Higher Derivatives](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM14-3-8.php)   [Differentials and Taylor Expansions](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM14-3-10.php)    Differentiability and the Chain Rule [Differentiability](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM14-5-2.php)   [The First Case of the Chain Rule](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM14-5-3.php)   [Chain Rule, General Case](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM14-5-4.php)   [Video: Worked problems](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM14-5-5.php)    Multiple Integrals [General Setup and Review of 1D Integrals](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM15-1-2.php)    [What is a Double 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of variables in 1 dimension](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM15-10-2.php)   [Mappings in 2 dimensions](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM15-10-3.php)   [Jacobians](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM15-10-4.php)   [Examples](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM15-10-5.php)   [Bonus: Cylindrical and spherical coordinates](https:\/\/web.ma.utexas.edu\/users\/m408s\/m408d\/CurrentWeb\/LM15-10-8.php) |   |   |\n|   |   |   |\n| **Notation for second partial derivatives** (fx)x\\=fxx\\=f11\\=∂∂x(∂f∂x)\\=∂2f∂x2\\=∂2z∂x2 ( f x ) x \\= f x x \\= f 11 \\= ∂ ∂ x ( ∂ f ∂ x ) \\= ∂ 2 f ∂ x 2 \\= ∂ 2 z ∂ x 2   (fx)y\\=fxy\\=f12\\=∂∂y(∂f∂x)\\=∂2f∂y∂x\\=∂2z∂y∂x ( f x ) y \\= f x y \\= f 12 \\= ∂ ∂ y ( ∂ f ∂ x ) \\= ∂ 2 f ∂ y ∂ x \\= ∂ 2 z ∂ y ∂ x   (fy)x\\=fyx\\=f21\\=∂∂x(∂f∂y)\\=∂2f∂x∂y\\=∂2z∂x∂y ( f y ) x \\= f y x \\= f 21 \\= ∂ ∂ x ( ∂ f ∂ y ) \\= ∂ 2 f ∂ x ∂ y \\= ∂ 2 z ∂ x ∂ y   (fy)y\\=fyy\\=f22\\=∂∂y(∂f∂y)\\=∂2f∂y2\\=∂2z∂y2 ( f y ) y \\= f y y \\= f 22 \\= ∂ ∂ y ( ∂ f ∂ y ) \\= ∂ 2 f ∂ y 2 \\= ∂ 2 z ∂ y 2 |   |   |\n|   |   |   |\n| **Clairaut's Theorem**   If fxy f x y and fyx f y x are both defined and continuous in a region containing the point (a,b) ( a , b ), then fxy(a,b)\\=fyx(a,b). f x y ( a , b ) \\= f y x ( a , b ) . |   |   |","attrs_readable_markdown":"### Higher Order Derivatives\nSince f x and f y are functions of x and y, **we can take derivatives of f x and f y to get second derivatives**. There are four such second derivatives, since each time we can differentiate with respect to x or y. Each of these second derivatives has multiple notations, and we have listed some of them.\n\n**Notation for second partial derivatives**\n\n( f x ) x \\= f x x \\= f 11 \\= ∂ ∂ x ( ∂ f ∂ x ) \\= ∂ 2 f ∂ x 2 \\= ∂ 2 z ∂ x 2\n\n( f x ) y \\= f x y \\= f 12 \\= ∂ ∂ y ( ∂ f ∂ x ) \\= ∂ 2 f ∂ y ∂ x \\= ∂ 2 z ∂ y ∂ x\n\n( f y ) x \\= f y x \\= f 21 \\= ∂ ∂ x ( ∂ f ∂ y ) \\= ∂ 2 f ∂ x ∂ y \\= ∂ 2 z ∂ x ∂ y\n\n( f y ) y \\= f y y \\= f 22 \\= ∂ ∂ y ( ∂ f ∂ y ) \\= ∂ 2 f ∂ y 2 \\= ∂ 2 z ∂ y 2\n\nA nice result regarding second partial derivatives is **Clairaut's Theorem**, which tells us that the mixed variable partial derivatives are equal.\n\n**Clairaut's Theorem**   \n If\n\nf\n\nx\n\ny\n\nand\n\nf\n\ny\n\nx\n\nare both defined and continuous in a region containing the point\n\n(\n\na\n\n,\n\nb\n\n)\n\n, then\n\nf x y ( a , b ) \\= f y x ( a , b ) .\n\nA consequence of this theorem is that we don't need to keep track of the order in which we take derivatives.\n\n**Example 1**: Let f ( x , y ) \\= 3 x 2 − 4 y 3 − 7 x 2 y 3. Previously, we determined that f x \\= 6 x − 14 x y 3 and f y \\= − 12 y 2 − 21 x 2 y 2. We have four second derivatives, but as Clairaut's Theorem tells us, f x y \\= f y x, so we really only need to compute three of them (we do all four to illustrate the theorem).\n\n**Solution 1**:  \n ( f x ) x \\= ∂ ∂ x f x \\= ∂ ∂ x ( 6 x − 14 x y 3 ) \\= 6 − 14 y 3 ( f x ) y \\= ∂ ∂ y f x \\= ∂ ∂ y ( 6 x − 14 x y 3 ) \\= 0 − 42 x y 2 \\= − 42 x y 2 ( f y ) x \\= ∂ ∂ x f y \\= ∂ ∂ x ( 12 y 2 − 21 x 2 y 2 ) \\= 0 − 42 x y 2 \\= − 42 x y 2 ( f y ) y \\= ∂ ∂ y f y \\= ∂ ∂ y ( 12 y 2 − 21 x 2 y 2 ) \\= 24 y − 42 x 2 y\n\nHigher partial derivatives and Clairaut's theorem are explained in the following video.","meta_canonical":null}

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Boilerpipe Text	Higher Order Derivatives Since f x and f y are functions of x and y , we can take derivatives of f x and f y to get second derivatives . There are four such second derivatives, since each time we can differentiate with respect to x or y . Each of these second derivatives has multiple notations, and we have listed some of them. Notation for second partial derivatives ( f x ) x = f x x = f 11 = ∂ ∂ x ( ∂ f ∂ x ) = ∂ 2 f ∂ x 2 = ∂ 2 z ∂ x 2 ( f x ) y = f x y = f 12 = ∂ ∂ y ( ∂ f ∂ x ) = ∂ 2 f ∂ y ∂ x = ∂ 2 z ∂ y ∂ x ( f y ) x = f y x = f 21 = ∂ ∂ x ( ∂ f ∂ y ) = ∂ 2 f ∂ x ∂ y = ∂ 2 z ∂ x ∂ y ( f y ) y = f y y = f 22 = ∂ ∂ y ( ∂ f ∂ y ) = ∂ 2 f ∂ y 2 = ∂ 2 z ∂ y 2 A nice result regarding second partial derivatives is Clairaut's Theorem , which tells us that the mixed variable partial derivatives are equal. Clairaut's Theorem If f x y and f y x are both defined and continuous in a region containing the point ( a , b ) , then f x y ( a , b ) = f y x ( a , b ) . A consequence of this theorem is that we don't need to keep track of the order in which we take derivatives. Example 1 : Let f ( x , y ) = 3 x 2 − 4 y 3 − 7 x 2 y 3 . Previously, we determined that f x = 6 x − 14 x y 3 and f y = − 12 y 2 − 21 x 2 y 2 . We have four second derivatives, but as Clairaut's Theorem tells us, f x y = f y x , so we really only need to compute three of them (we do all four to illustrate the theorem). Solution 1 : ( f x ) x = ∂ ∂ x f x = ∂ ∂ x ( 6 x − 14 x y 3 ) = 6 − 14 y 3 ( f x ) y = ∂ ∂ y f x = ∂ ∂ y ( 6 x − 14 x y 3 ) = 0 − 42 x y 2 = − 42 x y 2 ( f y ) x = ∂ ∂ x f y = ∂ ∂ x ( 12 y 2 − 21 x 2 y 2 ) = 0 − 42 x y 2 = − 42 x y 2 ( f y ) y = ∂ ∂ y f y = ∂ ∂ y ( 12 y 2 − 21 x 2 y 2 ) = 24 y − 42 x 2 y Higher partial derivatives and Clairaut's theorem are explained in the following video.
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Derivatives](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM14-3-8.php) [Differentials and Taylor Expansions](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM14-3-10.php) Differentiability and the Chain Rule [Differentiability](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM14-5-2.php) [The First Case of the Chain Rule](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM14-5-3.php) [Chain Rule, General Case](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM14-5-4.php) [Video: Worked problems](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM14-5-5.php) Multiple Integrals [General Setup and Review of 1D Integrals](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM15-1-2.php) [What is a Double Integral?](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM15-1-3.php) [Volumes as Double Integrals](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM15-1-4.php) Iterated Integrals over Rectangles [How To Compute Iterated Integrals](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM15-2-2.php) [Examples of Iterated Integrals](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM15-2-3.php) [Fubini's Theorem](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM15-2-7.php) [Summary and an Important Example](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM15-2-8.php) Double Integrals over General Regions [Type I and Type II regions](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM15-3-2.php) [Examples 1-4](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM15-3-3.php) [Examples 5-7](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM15-3-4.php) [Swapping the Order of Integration](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM15-3-6.php) [Area and Volume Revisited](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM15-3-7.php) Double integrals in polar coordinates [dA = r dr (d theta)](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM15-4-2.php) [Examples](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM15-4-3.php) Multiple integrals in physics [Double integrals in physics](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM15-5a-2.php) [Triple integrals in physics](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM15-5a-3.php) Integrals in Probability and Statistics [Single integrals in probability](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM15-5b-2.php) [Double integrals in probability](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM15-5b-3.php) Change of Variables [Review: Change of variables in 1 dimension](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM15-10-2.php) [Mappings in 2 dimensions](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM15-10-3.php) [Jacobians](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM15-10-4.php) [Examples](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM15-10-5.php) [Bonus: Cylindrical and spherical coordinates](https://web.ma.utexas.edu/users/m408s/m408d/CurrentWeb/LM15-10-8.php) \| \| \| \| \| \| \| \| Notation for second partial derivatives (fx)x\=fxx\=f11\=∂∂x(∂f∂x)\=∂2f∂x2\=∂2z∂x2 ( f x ) x \= f x x \= f 11 \= ∂ ∂ x ( ∂ f ∂ x ) \= ∂ 2 f ∂ x 2 \= ∂ 2 z ∂ x 2 (fx)y\=fxy\=f12\=∂∂y(∂f∂x)\=∂2f∂y∂x\=∂2z∂y∂x ( f x ) y \= f x y \= f 12 \= ∂ ∂ y ( ∂ f ∂ x ) \= ∂ 2 f ∂ y ∂ x \= ∂ 2 z ∂ y ∂ x (fy)x\=fyx\=f21\=∂∂x(∂f∂y)\=∂2f∂x∂y\=∂2z∂x∂y ( f y ) x \= f y x \= f 21 \= ∂ ∂ x ( ∂ f ∂ y ) \= ∂ 2 f ∂ x ∂ y \= ∂ 2 z ∂ x ∂ y (fy)y\=fyy\=f22\=∂∂y(∂f∂y)\=∂2f∂y2\=∂2z∂y2 ( f y ) y \= f y y \= f 22 \= ∂ ∂ y ( ∂ f ∂ y ) \= ∂ 2 f ∂ y 2 \= ∂ 2 z ∂ y 2 \| \| \| \| \| \| \| \| Clairaut's Theorem If fxy f x y and fyx f y x are both defined and continuous in a region containing the point (a,b) ( a , b ), then fxy(a,b)\=fyx(a,b). f x y ( a , b ) \= f y x ( a , b ) . \| \| \|
Readable Markdown	### Higher Order Derivatives Since f x and f y are functions of x and y, we can take derivatives of f x and f y to get second derivatives. There are four such second derivatives, since each time we can differentiate with respect to x or y. Each of these second derivatives has multiple notations, and we have listed some of them. Notation for second partial derivatives ( f x ) x \= f x x \= f 11 \= ∂ ∂ x ( ∂ f ∂ x ) \= ∂ 2 f ∂ x 2 \= ∂ 2 z ∂ x 2 ( f x ) y \= f x y \= f 12 \= ∂ ∂ y ( ∂ f ∂ x ) \= ∂ 2 f ∂ y ∂ x \= ∂ 2 z ∂ y ∂ x ( f y ) x \= f y x \= f 21 \= ∂ ∂ x ( ∂ f ∂ y ) \= ∂ 2 f ∂ x ∂ y \= ∂ 2 z ∂ x ∂ y ( f y ) y \= f y y \= f 22 \= ∂ ∂ y ( ∂ f ∂ y ) \= ∂ 2 f ∂ y 2 \= ∂ 2 z ∂ y 2 A nice result regarding second partial derivatives is Clairaut's Theorem, which tells us that the mixed variable partial derivatives are equal. Clairaut's Theorem If f x y and f y x are both defined and continuous in a region containing the point ( a , b ) , then f x y ( a , b ) \= f y x ( a , b ) . A consequence of this theorem is that we don't need to keep track of the order in which we take derivatives. Example 1: Let f ( x , y ) \= 3 x 2 − 4 y 3 − 7 x 2 y 3. Previously, we determined that f x \= 6 x − 14 x y 3 and f y \= − 12 y 2 − 21 x 2 y 2. We have four second derivatives, but as Clairaut's Theorem tells us, f x y \= f y x, so we really only need to compute three of them (we do all four to illustrate the theorem). Solution 1: ( f x ) x \= ∂ ∂ x f x \= ∂ ∂ x ( 6 x − 14 x y 3 ) \= 6 − 14 y 3 ( f x ) y \= ∂ ∂ y f x \= ∂ ∂ y ( 6 x − 14 x y 3 ) \= 0 − 42 x y 2 \= − 42 x y 2 ( f y ) x \= ∂ ∂ x f y \= ∂ ∂ x ( 12 y 2 − 21 x 2 y 2 ) \= 0 − 42 x y 2 \= − 42 x y 2 ( f y ) y \= ∂ ∂ y f y \= ∂ ∂ y ( 12 y 2 − 21 x 2 y 2 ) \= 24 y − 42 x 2 y Higher partial derivatives and Clairaut's theorem are explained in the following video.
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