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Calculated Shard: 16 (from laksa063)

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curl -X POST \
  'http://laksa016.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://tutorial.math.lamar.edu/classes/calciii/highorderpartialderivs.aspx\')), getAhrefsUnparsedNoserviceFromURL(\'https://tutorial.math.lamar.edu/classes/calciii/highorderpartialderivs.aspx\'))) FORMAT JSONEachRow'

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{"found_url":"https:\/\/tutorial.math.lamar.edu\/classes\/calciii\/highorderpartialderivs.aspx","crawl_time":1775432926,"first_indexed_time":1620385857,"http_code":200,"src_unparsed":"edu,lamar!math,tutorial,\/classes\/calciii\/highorderpartialderivs.aspx s443","src_root_hash":"6561424938415282416","history_drop_reason":null,"meta_title":"Calculus III - Higher Order Partial Derivatives","meta_descriptions":["In the section we will take a look at higher order partial derivatives. Unlike Calculus I however, we will have multiple second order derivatives, multiple third order derivatives, etc. because we are now working with functions of multiple variables. We will also discuss Clairaut’s Theorem to help with some of the work in finding higher order derivatives."],"attrs_boilerpipe_text":"Show Mobile Notice\nShow All Notes\n \nHide All Notes\nMobile Notice\nYou appear to be on a device with a \"narrow\" screen width (\ni.e.\nyou are probably on a mobile phone).  Due to the nature of the mathematics on this site it is best viewed in landscape mode.  If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll\/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.\nSection 13.4 : Higher Order Partial Derivatives\nJust as we had higher order derivatives with functions of one variable we will also have higher order derivatives of functions of more than one variable.  However, this time we will have more options since we do have more than one variable.\nConsider the case of a function of two variables, \\(f\\left( {x,y} \\right)\\) since both of the first order partial derivatives are also functions of \\(x\\) and \\(y\\) we could in turn differentiate each with respect to \\(x\\) or \\(y\\).  This means that for the case of a function of two variables there will be a total of four possible second order derivatives.  Here they are and the notations that we’ll use to denote them.\n\\[\\begin{align*}{\\left( {{f_x}} \\right)_x} & = {f_{x\\,x}} = \\frac{\\partial }{{\\partial x}}\\left( {\\frac{{\\partial f}}{{\\partial x}}} \\right) = \\frac{{{\\partial ^2}f}}{{\\partial {x^2}}}\\\\ {\\left( {{f_x}} \\right)_y} & = {f_{x\\,y}} = \\frac{\\partial }{{\\partial y}}\\left( {\\frac{{\\partial f}}{{\\partial x}}} \\right) = \\frac{{{\\partial ^2}f}}{{\\partial y\\partial x}}\\\\ {\\left( {{f_y}} \\right)_x} & = {f_{y\\,x}} = \\frac{\\partial }{{\\partial x}}\\left( {\\frac{{\\partial f}}{{\\partial y}}} \\right) = \\frac{{{\\partial ^2}f}}{{\\partial x\\partial y}}\\\\ {\\left( {{f_y}} \\right)_y} & = {f_{y\\,y}} = \\frac{\\partial }{{\\partial y}}\\left( {\\frac{{\\partial f}}{{\\partial y}}} \\right) = \\frac{{{\\partial ^2}f}}{{\\partial {y^2}}}\\end{align*}\\]\nThe second and third second order partial derivatives are often called mixed partial derivatives since we are taking derivatives with respect to more than one variable.  Note as well that the order that we take the derivatives in is given by the notation for each these.  If we are using the subscripting notation,\ne.g.\n\\({f_{x\\,y}}\\), then we will differentiate from left to right.  In other words, in this case, we will differentiate first with respect to \\(x\\) and then with respect to \\(y\\).  With the fractional notation,\ne.g.\n\\(\\frac{{{\\partial ^2}f}}{{\\partial y\\partial x}}\\), it is the opposite.  In these cases we differentiate moving along the denominator from right to left.  So, again, in this case we differentiate with respect to \\(x\\) first and then \\(y\\).\nLet’s take a quick look at an example.\nExample 1\nFind all the second order derivatives for \\(f\\left( {x,y} \\right) = \\cos \\left( {2x} \\right) - {x^2}{{\\bf{e}}^{5y}} + 3{y^2}\\).\nShow Solution\nWe’ll first need the first order derivatives so here they are.\n\\[\\begin{align*}{f_x}\\left( {x,y} \\right) & =  - 2\\sin \\left( {2x} \\right) - 2x{{\\bf{e}}^{5y}}\\\\ {f_y}\\left( {x,y} \\right) & =  - 5{x^2}{{\\bf{e}}^{5y}} + 6y\\end{align*}\\]\nNow, let’s get the second order derivatives.\n\\[\\begin{align*}{f_{xx}} & =  - 4\\cos \\left( {2x} \\right) - 2{{\\bf{e}}^{5y}}\\\\ {f_{xy}} & =  - 10x{{\\bf{e}}^{5y}}\\\\ {f_{yx}} & =  - 10x{{\\bf{e}}^{5y}}\\\\  {f_{yy}} & =  - 25{x^2}{{\\bf{e}}^{5y}} + 6\\end{align*}\\]\nNotice that we dropped the \\(\\left( {x,y} \\right)\\) from the derivatives.  This is fairly standard and we will be doing it most of the time from this point on.  We will also be dropping it for the first order derivatives in most cases.\nNow let’s also notice that, in this case, \\({f_{xy}} = {f_{yx}}\\).  This is not by coincidence.  If the function is “nice enough” this will always be the case.  So, what’s “nice enough”?  The following theorem tells us.\nClairaut’s Theorem\nSuppose that \\(f\\) is defined on a disk \\(D\\) that contains the point \\(\\left( {a,b} \\right)\\).  If the functions \\({f_{xy}}\\) and \\({f_{yx}}\\) are continuous on this disk then,\n\\[{f_{xy}}\\left( {a,b} \\right) = {f_{yx}}\\left( {a,b} \\right)\\]\nNow, do not get too excited about the disk business and the fact that we gave the theorem for a specific point.  In pretty much every example in this class if the two mixed second order partial derivatives are continuous then they will be equal.\nExample 2\nVerify Clairaut’s Theorem for \\(f\\left( {x,y} \\right) = x{{\\bf{e}}^{ - {x^2}{y^2}}}\\).\nShow Solution\nWe’ll first need the two first order derivatives.\n\\[\\begin{align*}{f_x}\\left( {x,y} \\right) & = {{\\bf{e}}^{ - {x^2}{y^2}}} - 2{x^2}{y^2}{{\\bf{e}}^{ - {x^2}{y^2}}}\\\\ {f_y}\\left( {x,y} \\right) & =  - 2y{x^3}{{\\bf{e}}^{ - {x^2}{y^2}}}\\end{align*}\\]\nNow, compute the two mixed second order partial derivatives.\n\\[\\begin{align*}{f_{xy}}\\left( {x,y} \\right) & =  - 2y{x^2}{{\\bf{e}}^{ - {x^2}{y^2}}} - 4{x^2}y{{\\bf{e}}^{ - {x^2}{y^2}}} + 4{x^4}{y^3}{{\\bf{e}}^{ - {x^2}{y^2}}} =  - 6{x^2}y{{\\bf{e}}^{ - {x^2}{y^2}}} + 4{x^4}{y^3}{{\\bf{e}}^{ - {x^2}{y^2}}}\\\\ {f_{yx}}\\left( {x,y} \\right) & =  - 6y{x^2}{{\\bf{e}}^{ - {x^2}{y^2}}} + 4{y^3}{x^4}{{\\bf{e}}^{ - {x^2}{y^2}}}\\end{align*}\\]\nSure enough they are the same.\nSo far we have only looked at second order derivatives.  There are, of course, higher order derivatives as well.  Here are a couple of the third order partial derivatives of function of two variables.\n\\[\\begin{align*}{f_{x\\,y\\,x}} & = {\\left( {{f_{xy}}} \\right)_x} = \\frac{\\partial }{{\\partial x}}\\left( {\\frac{{{\\partial ^2}f}}{{\\partial y\\partial x}}} \\right) = \\frac{{{\\partial ^3}f}}{{\\partial x\\partial y\\partial x}}\\\\ {f_{y\\,x\\,x}} & = {\\left( {{f_{yx}}} \\right)_x} = \\frac{\\partial }{{\\partial x}}\\left( {\\frac{{{\\partial ^2}f}}{{\\partial x\\partial y}}} \\right) = \\frac{{{\\partial ^3}f}}{{\\partial {x^2}\\partial y}}\\end{align*}\\]\nNotice as well that for both of these we differentiate once with respect to \\(y\\) and twice with respect to \\(x\\).  There is also another third order partial derivative in which we can do this, \\({f_{x\\,x\\,y}}\\).  There is an extension to Clairaut’s Theorem that says if all three of these are continuous then they should all be equal,\n\\[{f_{x\\,x\\,y}} = {f_{x\\,y\\,x}} = {f_{y\\,x\\,x}}\\]\nTo this point we’ve only looked at functions of two variables, but everything that we’ve done to this point will work regardless of the number of variables that we’ve got in the function and there are natural extensions to Clairaut’s theorem to all of these cases as well.  For instance,\n\\[{f_{x\\,z}}\\left( {x,y,z} \\right) = {f_{z\\,x}}\\left( {x,y,z} \\right)\\]\nprovided both of the derivatives are continuous.\nIn general, we can extend Clairaut’s theorem to any function and mixed partial derivatives.  The only requirement is that in each derivative we differentiate with respect to each variable the same number of times.  In other words, provided we meet the continuity condition, the following will be equal\n\\[{f_{s\\,s\\,r\\,t\\,s\\,r\\,r}} = {f_{t\\,r\\,s\\,r\\,s\\,s\\,r}}\\]\nbecause in each case we differentiate with respect to \\(t\\) once, \\(s\\) three times and \\(r\\) three times.\nLet’s do a couple of examples with higher (well higher order than two anyway) order derivatives and functions of more than two variables.\nExample 3\nFind the indicated derivative for each of the following functions.\nFind \\({f_{x\\,x\\,y\\,z\\,z}}\\) for \\(f\\left( {x,y,z} \\right) = {z^3}{y^2}\\ln \\left( x \\right)\\)\nFind \\(\\displaystyle \\frac{{{\\partial ^3}f}}{{\\partial y\\partial {x^2}}}\\) for \\(f\\left( {x,y} \\right) = {{\\bf{e}}^{xy}}\\)\nShow All Solutions\n \nHide All Solutions\na\nFind \\({f_{x\\,x\\,y\\,z\\,z}}\\) for \\(f\\left( {x,y,z} \\right) = {z^3}{y^2}\\ln \\left( x \\right)\\)\nShow Solution\nIn this case remember that we differentiate from left to right.  Here are the derivatives for this part.\n\\[{f_x} = \\frac{{{z^3}{y^2}}}{x}\\]\r\n          \r\n              \\[{f_{xx}} =  - \\frac{{{z^3}{y^2}}}{{{x^2}}}\\]\r\n          \r\n              \\[{f_{xxy}} =  - \\frac{{2{z^3}y}}{{{x^2}}}\\]\r\n          \r\n              \\[{f_{xxyz}} =  - \\frac{{6{z^2}y}}{{{x^2}}}\\]\r\n          \r\n              \\[{f_{xxyzz}} =  - \\frac{{12zy}}{{{x^2}}}\\]\nb\nFind \\(\\displaystyle \\frac{{{\\partial ^3}f}}{{\\partial y\\partial {x^2}}}\\) for \\(f\\left( {x,y} \\right) = {{\\bf{e}}^{xy}}\\)\nShow Solution\nHere we differentiate from right to left.  Here are the derivatives for this function.\n\\[\\frac{{\\partial f}}{{\\partial x}} = y{{\\bf{e}}^{xy}}\\]\r\n          \r\n              \\[\\frac{{{\\partial ^2}f}}{{\\partial {x^2}}} = {y^2}{{\\bf{e}}^{xy}}\\]\r\n          \r\n            \\[\\frac{{{\\partial ^3}f}}{{\\partial y\\partial {x^2}}} = 2y{{\\bf{e}}^{xy}} + x{y^2}{{\\bf{e}}^{xy}}\\]","attrs_markdown":"[Paul's Online Notes](https:\/\/tutorial.math.lamar.edu\/)\n\nNotes\n\nQuick Nav\n\nDownload\n\n- Go To\n- Notes\n- [Practice Problems](https:\/\/tutorial.math.lamar.edu\/Problems\/CalcIII\/HighOrderPartialDerivs.aspx \"Go to Practice Problems for current topic.\")\n- [Assignment Problems](https:\/\/tutorial.math.lamar.edu\/ProblemsNS\/CalcIII\/HighOrderPartialDerivs.aspx \"Go to Assignment Problems for current topic.\")\n- Show\/Hide\n- [Show all Solutions\/Steps\/*etc.*]()\n- [Hide all Solutions\/Steps\/*etc.*]()\n\n- Sections\n- [Interpretations of Partial Derivatives](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/PartialDerivInterp.aspx \"Previous Section : Interpretations of Partial Derivatives\")\n- [Differentials](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/Differentials.aspx \"Next Section : Differentials\")\n- Chapters\n- [3-Dimensional Space](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/3DSpace.aspx \"Previous Chapter : 3-Dimensional Space\")\n- [Applications of Partial Derivatives](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/PartialDerivAppsIntro.aspx \"Next Chapter : Applications of Partial Derivatives\")\n- Classes\n- [Algebra](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Alg.aspx \"Go To Algebra Notes\")\n- [Calculus I](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/CalcI.aspx \"Go To Calculus I Notes\")\n- [Calculus II](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/CalcII.aspx \"Go To Calculus II Notes\")\n- Calculus III\n- [Differential Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/DE.aspx \"Go To Differential Equations Notes\")\n- Extras\n- [Algebra & Trig Review](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/AlgebraTrig.aspx \"Go To Algebra & Trig Review\")\n- [Common Math Errors](https:\/\/tutorial.math.lamar.edu\/Extras\/CommonErrors\/CommonMathErrors.aspx \"Go To Common Math Errors\")\n- [Complex Number Primer](https:\/\/tutorial.math.lamar.edu\/Extras\/ComplexPrimer\/ComplexNumbers.aspx \"Go To Complex Numbers Primer\")\n- [How To Study Math](https:\/\/tutorial.math.lamar.edu\/Extras\/StudyMath\/HowToStudyMath.aspx \"Go To How To Study Math\")\n- [Cheat Sheets & Tables](https:\/\/tutorial.math.lamar.edu\/Extras\/CheatSheets_Tables.aspx \"Go To List of Cheat Sheets and Tables\")\n- Misc\n- [Contact Me](https:\/\/tutorial.math.lamar.edu\/contact.aspx \"Contact Me!\")\n- [MathJax Help and Configuration](https:\/\/tutorial.math.lamar.edu\/mathjax.aspx \"Info on MathJax and MathJax Configuration Menu\")\n\n- Notes Downloads\n- [Complete Book](https:\/\/tutorial.math.lamar.edu\/GetFile.aspx?file=B,20,N)\n- Practice Problems Downloads\n- [Complete Book - Problems Only](https:\/\/tutorial.math.lamar.edu\/GetFile.aspx?file=B,20,P)\n- [Complete Book - Solutions](https:\/\/tutorial.math.lamar.edu\/GetFile.aspx?file=B,20,S)\n- Assignment Problems Downloads\n- [Complete Book](https:\/\/tutorial.math.lamar.edu\/GetFile.aspx?file=B,20,A)\n- Other Items\n- [Get URL's for Download Items](https:\/\/tutorial.math.lamar.edu\/DownloadURLs.aspx?bi=20)\n\n- [Print Page in Current Form (Default)]()\n- [Show all Solutions\/Steps and Print Page]()\n- [Hide all Solutions\/Steps and Print Page]()\n\n- [Home](https:\/\/tutorial.math.lamar.edu\/)\n- Classes\n- [Algebra](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Alg.aspx)\n  - [1\\. Preliminaries](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Preliminaries.aspx)\n    - [1\\.1 Integer Exponents](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/IntegerExponents.aspx)\n    - [1\\.2 Rational Exponents](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/RationalExponents.aspx)\n    - [1\\.3 Radicals](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Radicals.aspx)\n    - [1\\.4 Polynomials](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Polynomials.aspx)\n    - [1\\.5 Factoring Polynomials](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Factoring.aspx)\n    - [1\\.6 Rational Expressions](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/RationalExpressions.aspx)\n    - [1\\.7 Complex Numbers](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/ComplexNumbers.aspx)\n  - [2\\. Solving Equations and Inequalities](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Solving.aspx)\n    - [2\\.1 Solutions and Solution Sets](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolutionSets.aspx)\n    - [2\\.2 Linear Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolveLinearEqns.aspx)\n    - [2\\.3 Applications of Linear Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/LinearApps.aspx)\n    - [2\\.4 Equations With More Than One Variable](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolveMultiVariable.aspx)\n    - [2\\.5 Quadratic Equations - Part I](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolveQuadraticEqnsI.aspx)\n    - [2\\.6 Quadratic Equations - Part II](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolveQuadraticEqnsII.aspx)\n    - [2\\.7 Quadratic Equations : A Summary](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolveQuadraticEqnSummary.aspx)\n    - [2\\.8 Applications of Quadratic Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/QuadraticApps.aspx)\n    - [2\\.9 Equations Reducible to Quadratic in Form](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/ReducibleToQuadratic.aspx)\n    - [2\\.10 Equations with Radicals](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolveRadicalEqns.aspx)\n    - [2\\.11 Linear Inequalities](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolveLinearInequalities.aspx)\n    - [2\\.12 Polynomial Inequalities](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolvePolyInequalities.aspx)\n    - [2\\.13 Rational Inequalities](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolveRationalInequalities.aspx)\n    - [2\\.14 Absolute Value Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolveAbsValueEqns.aspx)\n    - [2\\.15 Absolute Value Inequalities](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolveAbsValueIneq.aspx)\n  - [3\\. Graphing and Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Graphing_Functions.aspx)\n    - [3\\.1 Graphing](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Graphing.aspx)\n    - [3\\.2 Lines](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Lines.aspx)\n    - [3\\.3 Circles](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Circles.aspx)\n    - [3\\.4 The Definition of a Function](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/FunctionDefn.aspx)\n    - [3\\.5 Graphing Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/GraphFunctions.aspx)\n    - [3\\.6 Combining Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/CombineFunctions.aspx)\n    - [3\\.7 Inverse Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/InverseFunctions.aspx)\n  - [4\\. Common Graphs](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/CommonGraphs.aspx)\n    - [4\\.1 Lines, Circles and Piecewise Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Lines_Circles_PWF.aspx)\n    - [4\\.2 Parabolas](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Parabolas.aspx)\n    - [4\\.3 Ellipses](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Ellipses.aspx)\n    - [4\\.4 Hyperbolas](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Hyperbolas.aspx)\n    - [4\\.5 Miscellaneous Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/MiscFunctions.aspx)\n    - [4\\.6 Transformations](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Transformations.aspx)\n    - [4\\.7 Symmetry](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Symmetry.aspx)\n    - [4\\.8 Rational Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/GraphRationalFcns.aspx)\n  - [5\\. Polynomial Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/PolynomialFunctions.aspx)\n    - [5\\.1 Dividing Polynomials](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/DividingPolynomials.aspx)\n    - [5\\.2 Zeroes\/Roots of Polynomials](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/ZeroesOfPolynomials.aspx)\n    - [5\\.3 Graphing Polynomials](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/GraphingPolynomials.aspx)\n    - [5\\.4 Finding Zeroes of Polynomials](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/FindingZeroesOfPolynomials.aspx)\n    - [5\\.5 Partial Fractions](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/PartialFractions.aspx)\n  - [6\\. Exponential and Logarithm Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/ExpAndLog.aspx)\n    - [6\\.1 Exponential Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/ExpFunctions.aspx)\n    - [6\\.2 Logarithm Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/LogFunctions.aspx)\n    - [6\\.3 Solving Exponential Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolveExpEqns.aspx)\n    - [6\\.4 Solving Logarithm Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SolveLogEqns.aspx)\n    - [6\\.5 Applications](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/ExpLogApplications.aspx)\n  - [7\\. Systems of Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/Systems.aspx)\n    - [7\\.1 Linear Systems with Two Variables](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SystemsTwoVrble.aspx)\n    - [7\\.2 Linear Systems with Three Variables](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/SystemsThreeVrble.aspx)\n    - [7\\.3 Augmented Matrices](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/AugmentedMatrix.aspx)\n    - [7\\.4 More on the Augmented Matrix](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/AugmentedMatrixII.aspx)\n    - [7\\.5 Nonlinear Systems](https:\/\/tutorial.math.lamar.edu\/Classes\/Alg\/NonlinearSystems.aspx)\n- [Calculus I](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/CalcI.aspx)\n  - [1\\. Review](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ReviewIntro.aspx)\n    - [1\\.1 Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/Functions.aspx)\n    - [1\\.2 Inverse Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/InverseFunctions.aspx)\n    - [1\\.3 Trig Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/TrigFcns.aspx)\n    - [1\\.4 Solving Trig Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/TrigEquations.aspx)\n    - [1\\.5 Trig Equations with Calculators, Part I](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/TrigEquations_CalcI.aspx)\n    - [1\\.6 Trig Equations with Calculators, Part II](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/TrigEquations_CalcII.aspx)\n    - [1\\.7 Exponential Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ExpFunctions.aspx)\n    - [1\\.8 Logarithm Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/LogFcns.aspx)\n    - [1\\.9 Exponential and Logarithm Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ExpLogEqns.aspx)\n    - [1\\.10 Common Graphs](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/CommonGraphs.aspx)\n  - [2\\. Limits](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/limitsIntro.aspx)\n    - [2\\.1 Tangent Lines and Rates of Change](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/Tangents_Rates.aspx)\n    - [2\\.2 The Limit](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/TheLimit.aspx)\n    - [2\\.3 One-Sided Limits](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/OneSidedLimits.aspx)\n    - [2\\.4 Limit Properties](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/LimitsProperties.aspx)\n    - [2\\.5 Computing Limits](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ComputingLimits.aspx)\n    - [2\\.6 Infinite Limits](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/InfiniteLimits.aspx)\n    - [2\\.7 Limits At Infinity, Part I](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/LimitsAtInfinityI.aspx)\n    - [2\\.8 Limits At Infinity, Part II](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/LimitsAtInfinityII.aspx)\n    - [2\\.9 Continuity](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/Continuity.aspx)\n    - [2\\.10 The Definition of the Limit](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/DefnOfLimit.aspx)\n  - [3\\. Derivatives](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/DerivativeIntro.aspx)\n    - [3\\.1 The Definition of the Derivative](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/DefnOfDerivative.aspx)\n    - [3\\.2 Interpretation of the Derivative](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/DerivativeInterp.aspx)\n    - [3\\.3 Differentiation Formulas](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/DiffFormulas.aspx)\n    - [3\\.4 Product and Quotient Rule](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ProductQuotientRule.aspx)\n    - [3\\.5 Derivatives of Trig Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/DiffTrigFcns.aspx)\n    - [3\\.6 Derivatives of Exponential and Logarithm Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/DiffExpLogFcns.aspx)\n    - [3\\.7 Derivatives of Inverse Trig Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/DiffInvTrigFcns.aspx)\n    - [3\\.8 Derivatives of Hyperbolic Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/DiffHyperFcns.aspx)\n    - [3\\.9 Chain Rule](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ChainRule.aspx)\n    - [3\\.10 Implicit Differentiation](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ImplicitDIff.aspx)\n    - [3\\.11 Related Rates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/RelatedRates.aspx)\n    - [3\\.12 Higher Order Derivatives](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/HigherOrderDerivatives.aspx)\n    - [3\\.13 Logarithmic Differentiation](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/LogDiff.aspx)\n  - [4\\. Applications of Derivatives](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/DerivAppsIntro.aspx)\n    - [4\\.1 Rates of Change](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/RateOfChange.aspx)\n    - [4\\.2 Critical Points](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/CriticalPoints.aspx)\n    - [4\\.3 Minimum and Maximum Values](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/MinMaxValues.aspx)\n    - [4\\.4 Finding Absolute Extrema](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/AbsExtrema.aspx)\n    - [4\\.5 The Shape of a Graph, Part I](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ShapeofGraphPtI.aspx)\n    - [4\\.6 The Shape of a Graph, Part II](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ShapeofGraphPtII.aspx)\n    - [4\\.7 The Mean Value Theorem](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/MeanValueTheorem.aspx)\n    - [4\\.8 Optimization](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/Optimization.aspx)\n    - [4\\.9 More Optimization Problems](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/MoreOptimization.aspx)\n    - [4\\.10 L'Hospital's Rule and Indeterminate Forms](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/LHospitalsRule.aspx)\n    - [4\\.11 Linear Approximations](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/LinearApproximations.aspx)\n    - [4\\.12 Differentials](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/Differentials.aspx)\n    - [4\\.13 Newton's Method](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/NewtonsMethod.aspx)\n    - [4\\.14 Business Applications](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/BusinessApps.aspx)\n  - [5\\. Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/IntegralsIntro.aspx)\n    - [5\\.1 Indefinite Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/IndefiniteIntegrals.aspx)\n    - [5\\.2 Computing Indefinite Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ComputingIndefiniteIntegrals.aspx)\n    - [5\\.3 Substitution Rule for Indefinite Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/SubstitutionRuleIndefinite.aspx)\n    - [5\\.4 More Substitution Rule](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/SubstitutionRuleIndefinitePtII.aspx)\n    - [5\\.5 Area Problem](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/AreaProblem.aspx)\n    - [5\\.6 Definition of the Definite Integral](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/DefnOfDefiniteIntegral.aspx)\n    - [5\\.7 Computing Definite Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ComputingDefiniteIntegrals.aspx)\n    - [5\\.8 Substitution Rule for Definite Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/SubstitutionRuleDefinite.aspx)\n  - [6\\. Applications of Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/IntAppsIntro.aspx)\n    - [6\\.1 Average Function Value](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/AvgFcnValue.aspx)\n    - [6\\.2 Area Between Curves](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/AreaBetweenCurves.aspx)\n    - [6\\.3 Volumes of Solids of Revolution \/ Method of Rings](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/VolumeWithRings.aspx)\n    - [6\\.4 Volumes of Solids of Revolution\/Method of Cylinders](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/VolumeWithCylinder.aspx)\n    - [6\\.5 More Volume Problems](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/MoreVolume.aspx)\n    - [6\\.6 Work](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/Work.aspx)\n  - [Appendix A. Extras](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ExtrasIntro.aspx)\n    - [A.1 Proof of Various Limit Properties](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/LimitProofs.aspx)\n    - [A.2 Proof of Various Derivative Properties](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/DerivativeProofs.aspx)\n    - [A.3 Proof of Trig Limits](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ProofTrigDeriv.aspx)\n    - [A.4 Proofs of Derivative Applications Facts](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/DerivativeAppsProofs.aspx)\n    - [A.5 Proof of Various Integral Properties](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ProofIntProp.aspx)\n    - [A.6 Area and Volume Formulas](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/Area_Volume_Formulas.aspx)\n    - [A.7 Types of Infinity](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/TypesOfInfinity.aspx)\n    - [A.8 Summation Notation](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/SummationNotation.aspx)\n    - [A.9 Constant of Integration](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcI\/ConstantofIntegration.aspx)\n- [Calculus II](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/CalcII.aspx)\n  - [7\\. Integration Techniques](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/IntTechIntro.aspx)\n    - [7\\.1 Integration by Parts](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/IntegrationByParts.aspx)\n    - [7\\.2 Integrals Involving Trig Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/IntegralsWithTrig.aspx)\n    - [7\\.3 Trig Substitutions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/TrigSubstitutions.aspx)\n    - [7\\.4 Partial Fractions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/PartialFractions.aspx)\n    - [7\\.5 Integrals Involving Roots](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/IntegralsWithRoots.aspx)\n    - [7\\.6 Integrals Involving Quadratics](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/IntegralsWithQuadratics.aspx)\n    - [7\\.7 Integration Strategy](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/IntegrationStrategy.aspx)\n    - [7\\.8 Improper Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/ImproperIntegrals.aspx)\n    - [7\\.9 Comparison Test for Improper Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/ImproperIntegralsCompTest.aspx)\n    - [7\\.10 Approximating Definite Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/ApproximatingDefIntegrals.aspx)\n  - [8\\. Applications of Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/IntAppsIntro.aspx)\n    - [8\\.1 Arc Length](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/ArcLength.aspx)\n    - [8\\.2 Surface Area](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/SurfaceArea.aspx)\n    - [8\\.3 Center of Mass](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/CenterOfMass.aspx)\n    - [8\\.4 Hydrostatic Pressure](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/HydrostaticPressure.aspx)\n    - [8\\.5 Probability](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/Probability.aspx)\n  - [9\\. Parametric Equations and Polar Coordinates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/ParametricIntro.aspx)\n    - [9\\.1 Parametric Equations and Curves](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/ParametricEqn.aspx)\n    - [9\\.2 Tangents with Parametric Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/ParaTangent.aspx)\n    - [9\\.3 Area with Parametric Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/ParaArea.aspx)\n    - [9\\.4 Arc Length with Parametric Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/ParaArcLength.aspx)\n    - [9\\.5 Surface Area with Parametric Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/ParaSurfaceArea.aspx)\n    - [9\\.6 Polar Coordinates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/PolarCoordinates.aspx)\n    - [9\\.7 Tangents with Polar Coordinates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/PolarTangents.aspx)\n    - [9\\.8 Area with Polar Coordinates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/PolarArea.aspx)\n    - [9\\.9 Arc Length with Polar Coordinates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/PolarArcLength.aspx)\n    - [9\\.10 Surface Area with Polar Coordinates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/PolarSurfaceArea.aspx)\n    - [9\\.11 Arc Length and Surface Area Revisited](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/ArcLength_SurfaceArea.aspx)\n  - [10\\. Series & Sequences](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/SeriesIntro.aspx)\n    - [10\\.1 Sequences](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/Sequences.aspx)\n    - [10\\.2 More on Sequences](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/MoreSequences.aspx)\n    - [10\\.3 Series - The Basics](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/Series_Basics.aspx)\n    - [10\\.4 Convergence\/Divergence of Series](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/ConvergenceOfSeries.aspx)\n    - [10\\.5 Special Series](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/Series_Special.aspx)\n    - [10\\.6 Integral Test](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/IntegralTest.aspx)\n    - [10\\.7 Comparison Test\/Limit Comparison Test](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/SeriesCompTest.aspx)\n    - [10\\.8 Alternating Series Test](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/AlternatingSeries.aspx)\n    - [10\\.9 Absolute Convergence](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/AbsoluteConvergence.aspx)\n    - [10\\.10 Ratio Test](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/RatioTest.aspx)\n    - [10\\.11 Root Test](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/RootTest.aspx)\n    - [10\\.12 Strategy for Series](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/SeriesStrategy.aspx)\n    - [10\\.13 Estimating the Value of a Series](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/EstimatingSeries.aspx)\n    - [10\\.14 Power Series](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/PowerSeries.aspx)\n    - [10\\.15 Power Series and Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/PowerSeriesandFunctions.aspx)\n    - [10\\.16 Taylor Series](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/TaylorSeries.aspx)\n    - [10\\.17 Applications of Series](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/TaylorSeriesApps.aspx)\n    - [10\\.18 Binomial Series](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/BinomialSeries.aspx)\n  - [11\\. Vectors](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/VectorsIntro.aspx)\n    - [11\\.1 Vectors - The Basics](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/Vectors_Basics.aspx)\n    - [11\\.2 Vector Arithmetic](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/VectorArithmetic.aspx)\n    - [11\\.3 Dot Product](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/DotProduct.aspx)\n    - [11\\.4 Cross Product](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/CrossProduct.aspx)\n  - [12\\. 3-Dimensional Space](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/3DSpace.aspx)\n    - [12\\.1 The 3-D Coordinate System](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/3DCoords.aspx)\n    - [12\\.2 Equations of Lines](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/EqnsOfLines.aspx)\n    - [12\\.3 Equations of Planes](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/EqnsOfPlanes.aspx)\n    - [12\\.4 Quadric Surfaces](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/QuadricSurfaces.aspx)\n    - [12\\.5 Functions of Several Variables](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/MultiVrbleFcns.aspx)\n    - [12\\.6 Vector Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/VectorFunctions.aspx)\n    - [12\\.7 Calculus with Vector Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/VectorFcnsCalculus.aspx)\n    - [12\\.8 Tangent, Normal and Binormal Vectors](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/TangentNormalVectors.aspx)\n    - [12\\.9 Arc Length with Vector Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/VectorArcLength.aspx)\n    - [12\\.10 Curvature](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/Curvature.aspx)\n    - [12\\.11 Velocity and Acceleration](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/Velocity_Acceleration.aspx)\n    - [12\\.12 Cylindrical Coordinates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/CylindricalCoords.aspx)\n    - [12\\.13 Spherical Coordinates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/SphericalCoords.aspx)\n- [Calculus III](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/CalcIII.aspx)\n  - [12\\. 3-Dimensional Space](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/3DSpace.aspx)\n    - [12\\.1 The 3-D Coordinate System](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/3DCoords.aspx)\n    - [12\\.2 Equations of Lines](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/EqnsOfLines.aspx)\n    - [12\\.3 Equations of Planes](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/EqnsOfPlanes.aspx)\n    - [12\\.4 Quadric Surfaces](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/QuadricSurfaces.aspx)\n    - [12\\.5 Functions of Several Variables](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/MultiVrbleFcns.aspx)\n    - [12\\.6 Vector Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/VectorFunctions.aspx)\n    - [12\\.7 Calculus with Vector Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/VectorFcnsCalculus.aspx)\n    - [12\\.8 Tangent, Normal and Binormal Vectors](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/TangentNormalVectors.aspx)\n    - [12\\.9 Arc Length with Vector Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/VectorArcLength.aspx)\n    - [12\\.10 Curvature](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/Curvature.aspx)\n    - [12\\.11 Velocity and Acceleration](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/Velocity_Acceleration.aspx)\n    - [12\\.12 Cylindrical Coordinates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/CylindricalCoords.aspx)\n    - [12\\.13 Spherical Coordinates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/SphericalCoords.aspx)\n  - [13\\. Partial Derivatives](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/PartialDerivsIntro.aspx)\n    - [13\\.1 Limits](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/Limits.aspx)\n    - [13\\.2 Partial Derivatives](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/PartialDerivatives.aspx)\n    - [13\\.3 Interpretations of Partial Derivatives](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/PartialDerivInterp.aspx)\n    - [13\\.4 Higher Order Partial Derivatives](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/HighOrderPartialDerivs.aspx)\n    - [13\\.5 Differentials](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/Differentials.aspx)\n    - [13\\.6 Chain Rule](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/ChainRule.aspx)\n    - [13\\.7 Directional Derivatives](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/DirectionalDeriv.aspx)\n  - [14\\. Applications of Partial Derivatives](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/PartialDerivAppsIntro.aspx)\n    - [14\\.1 Tangent Planes and Linear Approximations](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/TangentPlanes.aspx)\n    - [14\\.2 Gradient Vector, Tangent Planes and Normal Lines](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/GradientVectorTangentPlane.aspx)\n    - [14\\.3 Relative Minimums and Maximums](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/RelativeExtrema.aspx)\n    - [14\\.4 Absolute Minimums and Maximums](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/AbsoluteExtrema.aspx)\n    - [14\\.5 Lagrange Multipliers](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/LagrangeMultipliers.aspx)\n  - [15\\. Multiple Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/MultipleIntegralsIntro.aspx)\n    - [15\\.1 Double Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/DoubleIntegrals.aspx)\n    - [15\\.2 Iterated Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/IteratedIntegrals.aspx)\n    - [15\\.3 Double Integrals over General Regions](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/DIGeneralRegion.aspx)\n    - [15\\.4 Double Integrals in Polar Coordinates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/DIPolarCoords.aspx)\n    - [15\\.5 Triple Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/TripleIntegrals.aspx)\n    - [15\\.6 Triple Integrals in Cylindrical Coordinates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/TICylindricalCoords.aspx)\n    - [15\\.7 Triple Integrals in Spherical Coordinates](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/TISphericalCoords.aspx)\n    - [15\\.8 Change of Variables](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/ChangeOfVariables.aspx)\n    - [15\\.9 Surface Area](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/SurfaceArea.aspx)\n    - [15\\.10 Area and Volume Revisited](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/Area_Volume.aspx)\n  - [16\\. Line Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/LineIntegralsIntro.aspx)\n    - [16\\.1 Vector Fields](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/VectorFields.aspx)\n    - [16\\.2 Line Integrals - Part I](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/LineIntegralsPtI.aspx)\n    - [16\\.3 Line Integrals - Part II](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/LineIntegralsPtII.aspx)\n    - [16\\.4 Line Integrals of Vector Fields](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/LineIntegralsVectorFields.aspx)\n    - [16\\.5 Fundamental Theorem for Line Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/FundThmLineIntegrals.aspx)\n    - [16\\.6 Conservative Vector Fields](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/ConservativeVectorField.aspx)\n    - [16\\.7 Green's Theorem](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/GreensTheorem.aspx)\n  - [17\\.Surface Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/SurfaceIntegralsIntro.aspx)\n    - [17\\.1 Curl and Divergence](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/CurlDivergence.aspx)\n    - [17\\.2 Parametric Surfaces](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/ParametricSurfaces.aspx)\n    - [17\\.3 Surface Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/SurfaceIntegrals.aspx)\n    - [17\\.4 Surface Integrals of Vector Fields](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/SurfIntVectorField.aspx)\n    - [17\\.5 Stokes' Theorem](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/StokesTheorem.aspx)\n    - [17\\.6 Divergence Theorem](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/DivergenceTheorem.aspx)\n- [Differential Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/DE.aspx)\n  - [1\\. Basic Concepts](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/IntroBasic.aspx)\n    - [1\\.1 Definitions](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/Definitions.aspx)\n    - [1\\.2 Direction Fields](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/DirectionFields.aspx)\n    - [1\\.3 Final Thoughts](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/FinalThoughts.aspx)\n  - [2\\. First Order DE's](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/IntroFirstOrder.aspx)\n    - [2\\.1 Linear Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/Linear.aspx)\n    - [2\\.2 Separable Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/Separable.aspx)\n    - [2\\.3 Exact Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/Exact.aspx)\n    - [2\\.4 Bernoulli Differential Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/Bernoulli.aspx)\n    - [2\\.5 Substitutions](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/Substitutions.aspx)\n    - [2\\.6 Intervals of Validity](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/IoV.aspx)\n    - [2\\.7 Modeling with First Order DE's](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/Modeling.aspx)\n    - [2\\.8 Equilibrium Solutions](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/EquilibriumSolutions.aspx)\n    - [2\\.9 Euler's Method](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/EulersMethod.aspx)\n  - [3\\. Second Order DE's](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/IntroSecondOrder.aspx)\n    - [3\\.1 Basic Concepts](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/SecondOrderConcepts.aspx)\n    - [3\\.2 Real & Distinct Roots](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/RealRoots.aspx)\n    - [3\\.3 Complex Roots](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/ComplexRoots.aspx)\n    - [3\\.4 Repeated Roots](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/RepeatedRoots.aspx)\n    - [3\\.5 Reduction of Order](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/ReductionofOrder.aspx)\n    - [3\\.6 Fundamental Sets of Solutions](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/FundamentalSetsofSolutions.aspx)\n    - [3\\.7 More on the Wronskian](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/Wronskian.aspx)\n    - [3\\.8 Nonhomogeneous Differential Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/NonhomogeneousDE.aspx)\n    - [3\\.9 Undetermined Coefficients](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/UndeterminedCoefficients.aspx)\n    - [3\\.10 Variation of Parameters](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/VariationofParameters.aspx)\n    - [3\\.11 Mechanical Vibrations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/Vibrations.aspx)\n  - [4\\. Laplace Transforms](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/LaplaceIntro.aspx)\n    - [4\\.1 The Definition](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/LaplaceDefinition.aspx)\n    - [4\\.2 Laplace Transforms](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/LaplaceTransforms.aspx)\n    - [4\\.3 Inverse Laplace Transforms](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/InverseTransforms.aspx)\n    - [4\\.4 Step Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/StepFunctions.aspx)\n    - [4\\.5 Solving IVP's with Laplace Transforms](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/IVPWithLaplace.aspx)\n    - [4\\.6 Nonconstant Coefficient IVP's](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/IVPWithNonConstantCoefficient.aspx)\n    - [4\\.7 IVP's With Step Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/IVPWithStepFunction.aspx)\n    - [4\\.8 Dirac Delta Function](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/DiracDeltaFunction.aspx)\n    - [4\\.9 Convolution Integrals](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/ConvolutionIntegrals.aspx)\n    - [4\\.10 Table Of Laplace Transforms](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/Laplace_Table.aspx)\n  - [5\\. Systems of DE's](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/SystemsIntro.aspx)\n    - [5\\.1 Review : Systems of Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/LA_Systems.aspx)\n    - [5\\.2 Review : Matrices & Vectors](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/LA_Matrix.aspx)\n    - [5\\.3 Review : Eigenvalues & Eigenvectors](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/LA_Eigen.aspx)\n    - [5\\.4 Systems of Differential Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/SystemsDE.aspx)\n    - [5\\.5 Solutions to Systems](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/SolutionsToSystems.aspx)\n    - [5\\.6 Phase Plane](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/PhasePlane.aspx)\n    - [5\\.7 Real Eigenvalues](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/RealEigenvalues.aspx)\n    - [5\\.8 Complex Eigenvalues](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/ComplexEigenvalues.aspx)\n    - [5\\.9 Repeated Eigenvalues](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/RepeatedEigenvalues.aspx)\n    - [5\\.10 Nonhomogeneous Systems](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/NonhomogeneousSystems.aspx)\n    - [5\\.11 Laplace Transforms](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/SystemsLaplace.aspx)\n    - [5\\.12 Modeling](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/SystemsModeling.aspx)\n  - [6\\. Series Solutions to DE's](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/SeriesIntro.aspx)\n    - [6\\.1 Review : Power Series](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/PowerSeries.aspx)\n    - [6\\.2 Review : Taylor Series](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/TaylorSeries.aspx)\n    - [6\\.3 Series Solutions](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/SeriesSolutions.aspx)\n    - [6\\.4 Euler Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/EulerEquations.aspx)\n  - [7\\. Higher Order Differential Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/IntroHigherOrder.aspx)\n    - [7\\.1 Basic Concepts for *n*th Order Linear Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/HOBasicConcepts.aspx)\n    - [7\\.2 Linear Homogeneous Differential Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/HOHomogeneousDE.aspx)\n    - [7\\.3 Undetermined Coefficients](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/HOUndeterminedCoeff.aspx)\n    - [7\\.4 Variation of Parameters](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/HOVariationOfParam.aspx)\n    - [7\\.5 Laplace Transforms](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/HOLaplaceTransforms.aspx)\n    - [7\\.6 Systems of Differential Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/HOSystems.aspx)\n    - [7\\.7 Series Solutions](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/HOSeries.aspx)\n  - [8\\. Boundary Value Problems & Fourier Series](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/IntroBVP.aspx)\n    - [8\\.1 Boundary Value Problems](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/BoundaryValueProblem.aspx)\n    - [8\\.2 Eigenvalues and Eigenfunctions](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/BVPEvals.aspx)\n    - [8\\.3 Periodic Functions & Orthogonal Functions](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/PeriodicOrthogonal.aspx)\n    - [8\\.4 Fourier Sine Series](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/FourierSineSeries.aspx)\n    - [8\\.5 Fourier Cosine Series](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/FourierCosineSeries.aspx)\n    - [8\\.6 Fourier Series](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/FourierSeries.aspx)\n    - [8\\.7 Convergence of Fourier Series](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/ConvergenceFourierSeries.aspx)\n  - [9\\. Partial Differential Equations](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/IntroPDE.aspx)\n    - [9\\.1 The Heat Equation](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/TheHeatEquation.aspx)\n    - [9\\.2 The Wave Equation](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/TheWaveEquation.aspx)\n    - [9\\.3 Terminology](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/PDETerminology.aspx)\n    - [9\\.4 Separation of Variables](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/SeparationofVariables.aspx)\n    - [9\\.5 Solving the Heat Equation](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/SolvingHeatEquation.aspx)\n    - [9\\.6 Heat Equation with Non-Zero Temperature Boundaries](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/HeatEqnNonZero.aspx)\n    - [9\\.7 Laplace's Equation](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/LaplacesEqn.aspx)\n    - [9\\.8 Vibrating String](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/VibratingString.aspx)\n    - [9\\.9 Summary of Separation of Variables](https:\/\/tutorial.math.lamar.edu\/Classes\/DE\/PDESummary.aspx)\n- Extras\n- [Algebra & Trig Review](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/AlgebraTrig.aspx)\n  - [1\\. Algebra](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/AlgebraIntro.aspx)\n    - [1\\.1 Exponents](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/Exponents.aspx)\n    - [1\\.2 Absolute Value](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/AbsoluteValue.aspx)\n    - [1\\.3 Radicals](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/Radicals.aspx)\n    - [1\\.4 Rationalizing](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/Rationalizing.aspx)\n    - [1\\.5 Functions](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/Functions.aspx)\n    - [1\\.6 Multiplying Polynomials](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/MultPoly.aspx)\n    - [1\\.7 Factoring](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/Factoring.aspx)\n    - [1\\.8 Simplifying Rational Expressions](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/SimpRatExp.aspx)\n    - [1\\.9 Graphing and Common Graphs](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/Graphing.aspx)\n    - [1\\.10 Solving Equations, Part I](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/SolveEqnPtI.aspx)\n    - [1\\.11 Solving Equations, Part II](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/SolveEqnPtII.aspx)\n    - [1\\.12 Solving Systems of Equations](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/SolveSystems.aspx)\n    - [1\\.13 Solving Inequalities](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/SolveIneq.aspx)\n    - [1\\.14 Absolute Value Equations and Inequalities](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/SolveAbsValue.aspx)\n  - [2\\. Trigonometry](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/TrigIntro.aspx)\n    - [2\\.1 Trig Function Evaluation](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/TrigFunctions.aspx)\n    - [2\\.2 Graphs of Trig Functions](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/TrigGraphs.aspx)\n    - [2\\.3 Trig Formulas](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/TrigFormulas.aspx)\n    - [2\\.4 Solving Trig Equations](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/SolveTrigEqn.aspx)\n    - [2\\.5 Inverse Trig Functions](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/InverseTrig.aspx)\n  - [3\\. Exponentials & Logarithms](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/ExpLogIntro.aspx)\n    - [3\\.1 Basic Exponential Functions](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/ExponentialFcns.aspx)\n    - [3\\.2 Basic Logarithm Functions](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/LogarithmFcns.aspx)\n    - [3\\.3 Logarithm Properties](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/LogProperties.aspx)\n    - [3\\.4 Simplifying Logarithms](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/SimpLogs.aspx)\n    - [3\\.5 Solving Exponential Equations](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/SolveExpEqn.aspx)\n    - [3\\.6 Solving Logarithm Equations](https:\/\/tutorial.math.lamar.edu\/Extras\/AlgebraTrigReview\/SolveLogEqn.aspx)\n- [Common Math Errors](https:\/\/tutorial.math.lamar.edu\/Extras\/CommonErrors\/CommonMathErrors.aspx)\n  - [1\\. General Errors](https:\/\/tutorial.math.lamar.edu\/Extras\/CommonErrors\/GeneralErrors.aspx)\n  - [2\\. Algebra Errors](https:\/\/tutorial.math.lamar.edu\/Extras\/CommonErrors\/AlgebraErrors.aspx)\n  - [3\\. Trig Errors](https:\/\/tutorial.math.lamar.edu\/Extras\/CommonErrors\/TrigErrors.aspx)\n  - [4\\. Common Errors](https:\/\/tutorial.math.lamar.edu\/Extras\/CommonErrors\/CommonErrors.aspx)\n  - [5\\. Calculus Errors](https:\/\/tutorial.math.lamar.edu\/Extras\/CommonErrors\/CalculusErrors.aspx)\n- [Complex Number Primer](https:\/\/tutorial.math.lamar.edu\/Extras\/ComplexPrimer\/ComplexNumbers.aspx)\n  - [1\\. The Definition](https:\/\/tutorial.math.lamar.edu\/Extras\/ComplexPrimer\/Definition.aspx)\n  - [2\\. Arithmetic](https:\/\/tutorial.math.lamar.edu\/Extras\/ComplexPrimer\/Arithmetic.aspx)\n  - [3\\. Conjugate and Modulus](https:\/\/tutorial.math.lamar.edu\/Extras\/ComplexPrimer\/ConjugateModulus.aspx)\n  - [4\\. Polar and Exponential Forms](https:\/\/tutorial.math.lamar.edu\/Extras\/ComplexPrimer\/Forms.aspx)\n  - [5\\. Powers and Roots](https:\/\/tutorial.math.lamar.edu\/Extras\/ComplexPrimer\/Roots.aspx)\n- [How To Study Math](https:\/\/tutorial.math.lamar.edu\/Extras\/StudyMath\/HowToStudyMath.aspx)\n  - [1\\. General Tips](https:\/\/tutorial.math.lamar.edu\/Extras\/StudyMath\/GeneralTips.aspx)\n  - [2\\. Taking Notes](https:\/\/tutorial.math.lamar.edu\/Extras\/StudyMath\/TakingNotes.aspx)\n  - [3\\. Getting Help](https:\/\/tutorial.math.lamar.edu\/Extras\/StudyMath\/GettingHelp.aspx)\n  - [4\\. Doing Homework](https:\/\/tutorial.math.lamar.edu\/Extras\/StudyMath\/Homework.aspx)\n  - [5\\. Problem Solving](https:\/\/tutorial.math.lamar.edu\/Extras\/StudyMath\/ProblemSolving.aspx)\n  - [6\\. Studying For an Exam](https:\/\/tutorial.math.lamar.edu\/Extras\/StudyMath\/StudyForExam.aspx)\n  - [7\\. Taking an Exam](https:\/\/tutorial.math.lamar.edu\/Extras\/StudyMath\/TakingExam.aspx)\n  - [8\\. Learn From Your Errors](https:\/\/tutorial.math.lamar.edu\/Extras\/StudyMath\/Errors.aspx)\n- [Cheat Sheets & Tables](https:\/\/tutorial.math.lamar.edu\/Extras\/CheatSheets_Tables.aspx)\n- Misc Links\n- [Contact Me](https:\/\/tutorial.math.lamar.edu\/contact.aspx)\n- [Links](https:\/\/tutorial.math.lamar.edu\/links.aspx)\n- [MathJax Help and Configuration](https:\/\/tutorial.math.lamar.edu\/mathjax.aspx)\n- [Privacy Statement](https:\/\/tutorial.math.lamar.edu\/privacy.aspx)\n- [Site Help & FAQ](https:\/\/tutorial.math.lamar.edu\/help.aspx)\n- [Terms of Use](https:\/\/tutorial.math.lamar.edu\/terms.aspx)\n\nPaul's Online Notes   \n  [Home](https:\/\/tutorial.math.lamar.edu\/ \"Go to Main Page\") \/  [Calculus III](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/CalcIII.aspx \"Go to Book Introduction\") \/  [Partial Derivatives](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/PartialDerivsIntro.aspx \"Go to Chapter Introduction\")  \/ Higher Order Partial Derivatives\n\n[Prev. Section](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/PartialDerivInterp.aspx \"Goto Previous Section : Interpretations of Partial Derivatives\")\n\nNotes [Practice Problems](https:\/\/tutorial.math.lamar.edu\/Problems\/CalcIII\/HighOrderPartialDerivs.aspx \"Go to Practice Problems for current topic.\") [Assignment Problems](https:\/\/tutorial.math.lamar.edu\/ProblemsNS\/CalcIII\/HighOrderPartialDerivs.aspx \"Go to Assignment Problems for current topic.\")\n\n[Next Section](https:\/\/tutorial.math.lamar.edu\/Classes\/CalcIII\/Differentials.aspx \"Goto Next Section : Differentials\")\n\nShow Mobile Notice  Show All Notes Hide All Notes\n\nMobile Notice\n\nYou appear to be on a device with a \"narrow\" screen width (*i.e.* you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll\/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.\n### Section 13.4 : Higher Order Partial Derivatives\nJust as we had higher order derivatives with functions of one variable we will also have higher order derivatives of functions of more than one variable. However, this time we will have more options since we do have more than one variable.\n\nConsider the case of a function of two variables, \\\\(f\\\\left( {x,y} \\\\right)\\\\) since both of the first order partial derivatives are also functions of \\\\(x\\\\) and \\\\(y\\\\) we could in turn differentiate each with respect to \\\\(x\\\\) or \\\\(y\\\\). This means that for the case of a function of two variables there will be a total of four possible second order derivatives. Here they are and the notations that we’ll use to denote them.\n\n\\\\\\[\\\\begin{align\\*}{\\\\left( {{f\\_x}} \\\\right)\\_x} & = {f\\_{x\\\\,x}} = \\\\frac{\\\\partial }{{\\\\partial x}}\\\\left( {\\\\frac{{\\\\partial f}}{{\\\\partial x}}} \\\\right) = \\\\frac{{{\\\\partial ^2}f}}{{\\\\partial {x^2}}}\\\\\\\\ {\\\\left( {{f\\_x}} \\\\right)\\_y} & = {f\\_{x\\\\,y}} = \\\\frac{\\\\partial }{{\\\\partial y}}\\\\left( {\\\\frac{{\\\\partial f}}{{\\\\partial x}}} \\\\right) = \\\\frac{{{\\\\partial ^2}f}}{{\\\\partial y\\\\partial x}}\\\\\\\\ {\\\\left( {{f\\_y}} \\\\right)\\_x} & = {f\\_{y\\\\,x}} = \\\\frac{\\\\partial }{{\\\\partial x}}\\\\left( {\\\\frac{{\\\\partial f}}{{\\\\partial y}}} \\\\right) = \\\\frac{{{\\\\partial ^2}f}}{{\\\\partial x\\\\partial y}}\\\\\\\\ {\\\\left( {{f\\_y}} \\\\right)\\_y} & = {f\\_{y\\\\,y}} = \\\\frac{\\\\partial }{{\\\\partial y}}\\\\left( {\\\\frac{{\\\\partial f}}{{\\\\partial y}}} \\\\right) = \\\\frac{{{\\\\partial ^2}f}}{{\\\\partial {y^2}}}\\\\end{align\\*}\\\\\\]\n\nThe second and third second order partial derivatives are often called mixed partial derivatives since we are taking derivatives with respect to more than one variable. Note as well that the order that we take the derivatives in is given by the notation for each these. If we are using the subscripting notation, *e.g.* \\\\({f\\_{x\\\\,y}}\\\\), then we will differentiate from left to right. In other words, in this case, we will differentiate first with respect to \\\\(x\\\\) and then with respect to \\\\(y\\\\). With the fractional notation, *e.g.* \\\\(\\\\frac{{{\\\\partial ^2}f}}{{\\\\partial y\\\\partial x}}\\\\), it is the opposite. In these cases we differentiate moving along the denominator from right to left. So, again, in this case we differentiate with respect to \\\\(x\\\\) first and then \\\\(y\\\\).\n\nLet’s take a quick look at an example.\n\nExample 1 Find all the second order derivatives for \\\\(f\\\\left( {x,y} \\\\right) = \\\\cos \\\\left( {2x} \\\\right) - {x^2}{{\\\\bf{e}}^{5y}} + 3{y^2}\\\\).\n\nShow Solution\n\nWe’ll first need the first order derivatives so here they are.\n\n\\\\\\[\\\\begin{align\\*}{f\\_x}\\\\left( {x,y} \\\\right) & = - 2\\\\sin \\\\left( {2x} \\\\right) - 2x{{\\\\bf{e}}^{5y}}\\\\\\\\ {f\\_y}\\\\left( {x,y} \\\\right) & = - 5{x^2}{{\\\\bf{e}}^{5y}} + 6y\\\\end{align\\*}\\\\\\]\n\nNow, let’s get the second order derivatives.\n\n\\\\\\[\\\\begin{align\\*}{f\\_{xx}} & = - 4\\\\cos \\\\left( {2x} \\\\right) - 2{{\\\\bf{e}}^{5y}}\\\\\\\\ {f\\_{xy}} & = - 10x{{\\\\bf{e}}^{5y}}\\\\\\\\ {f\\_{yx}} & = - 10x{{\\\\bf{e}}^{5y}}\\\\\\\\ {f\\_{yy}} & = - 25{x^2}{{\\\\bf{e}}^{5y}} + 6\\\\end{align\\*}\\\\\\]\n\nNotice that we dropped the \\\\(\\\\left( {x,y} \\\\right)\\\\) from the derivatives. This is fairly standard and we will be doing it most of the time from this point on. We will also be dropping it for the first order derivatives in most cases.\n\nNow let’s also notice that, in this case, \\\\({f\\_{xy}} = {f\\_{yx}}\\\\). This is not by coincidence. If the function is “nice enough” this will always be the case. So, what’s “nice enough”? The following theorem tells us.\n\n#### Clairaut’s Theorem\nSuppose that \\\\(f\\\\) is defined on a disk \\\\(D\\\\) that contains the point \\\\(\\\\left( {a,b} \\\\right)\\\\). If the functions \\\\({f\\_{xy}}\\\\) and \\\\({f\\_{yx}}\\\\) are continuous on this disk then,\n\n\\\\\\[{f\\_{xy}}\\\\left( {a,b} \\\\right) = {f\\_{yx}}\\\\left( {a,b} \\\\right)\\\\\\]\n\nNow, do not get too excited about the disk business and the fact that we gave the theorem for a specific point. In pretty much every example in this class if the two mixed second order partial derivatives are continuous then they will be equal.\n\nExample 2 Verify Clairaut’s Theorem for \\\\(f\\\\left( {x,y} \\\\right) = x{{\\\\bf{e}}^{ - {x^2}{y^2}}}\\\\).\n\nShow Solution\n\nWe’ll first need the two first order derivatives.\n\n\\\\\\[\\\\begin{align\\*}{f\\_x}\\\\left( {x,y} \\\\right) & = {{\\\\bf{e}}^{ - {x^2}{y^2}}} - 2{x^2}{y^2}{{\\\\bf{e}}^{ - {x^2}{y^2}}}\\\\\\\\ {f\\_y}\\\\left( {x,y} \\\\right) & = - 2y{x^3}{{\\\\bf{e}}^{ - {x^2}{y^2}}}\\\\end{align\\*}\\\\\\]\n\nNow, compute the two mixed second order partial derivatives.\n\n\\\\\\[\\\\begin{align\\*}{f\\_{xy}}\\\\left( {x,y} \\\\right) & = - 2y{x^2}{{\\\\bf{e}}^{ - {x^2}{y^2}}} - 4{x^2}y{{\\\\bf{e}}^{ - {x^2}{y^2}}} + 4{x^4}{y^3}{{\\\\bf{e}}^{ - {x^2}{y^2}}} = - 6{x^2}y{{\\\\bf{e}}^{ - {x^2}{y^2}}} + 4{x^4}{y^3}{{\\\\bf{e}}^{ - {x^2}{y^2}}}\\\\\\\\ {f\\_{yx}}\\\\left( {x,y} \\\\right) & = - 6y{x^2}{{\\\\bf{e}}^{ - {x^2}{y^2}}} + 4{y^3}{x^4}{{\\\\bf{e}}^{ - {x^2}{y^2}}}\\\\end{align\\*}\\\\\\]\n\nSure enough they are the same.\n\nSo far we have only looked at second order derivatives. There are, of course, higher order derivatives as well. Here are a couple of the third order partial derivatives of function of two variables.\n\n\\\\\\[\\\\begin{align\\*}{f\\_{x\\\\,y\\\\,x}} & = {\\\\left( {{f\\_{xy}}} \\\\right)\\_x} = \\\\frac{\\\\partial }{{\\\\partial x}}\\\\left( {\\\\frac{{{\\\\partial ^2}f}}{{\\\\partial y\\\\partial x}}} \\\\right) = \\\\frac{{{\\\\partial ^3}f}}{{\\\\partial x\\\\partial y\\\\partial x}}\\\\\\\\ {f\\_{y\\\\,x\\\\,x}} & = {\\\\left( {{f\\_{yx}}} \\\\right)\\_x} = \\\\frac{\\\\partial }{{\\\\partial x}}\\\\left( {\\\\frac{{{\\\\partial ^2}f}}{{\\\\partial x\\\\partial y}}} \\\\right) = \\\\frac{{{\\\\partial ^3}f}}{{\\\\partial {x^2}\\\\partial y}}\\\\end{align\\*}\\\\\\]\n\nNotice as well that for both of these we differentiate once with respect to \\\\(y\\\\) and twice with respect to \\\\(x\\\\). There is also another third order partial derivative in which we can do this, \\\\({f\\_{x\\\\,x\\\\,y}}\\\\). There is an extension to Clairaut’s Theorem that says if all three of these are continuous then they should all be equal,\n\n\\\\\\[{f\\_{x\\\\,x\\\\,y}} = {f\\_{x\\\\,y\\\\,x}} = {f\\_{y\\\\,x\\\\,x}}\\\\\\]\n\nTo this point we’ve only looked at functions of two variables, but everything that we’ve done to this point will work regardless of the number of variables that we’ve got in the function and there are natural extensions to Clairaut’s theorem to all of these cases as well. For instance,\n\n\\\\\\[{f\\_{x\\\\,z}}\\\\left( {x,y,z} \\\\right) = {f\\_{z\\\\,x}}\\\\left( {x,y,z} \\\\right)\\\\\\]\n\nprovided both of the derivatives are continuous.\n\nIn general, we can extend Clairaut’s theorem to any function and mixed partial derivatives. The only requirement is that in each derivative we differentiate with respect to each variable the same number of times. In other words, provided we meet the continuity condition, the following will be equal\n\n\\\\\\[{f\\_{s\\\\,s\\\\,r\\\\,t\\\\,s\\\\,r\\\\,r}} = {f\\_{t\\\\,r\\\\,s\\\\,r\\\\,s\\\\,s\\\\,r}}\\\\\\]\n\nbecause in each case we differentiate with respect to \\\\(t\\\\) once, \\\\(s\\\\) three times and \\\\(r\\\\) three times.\n\nLet’s do a couple of examples with higher (well higher order than two anyway) order derivatives and functions of more than two variables.\n\nExample 3 Find the indicated derivative for each of the following functions.\n\n1. Find \\\\({f\\_{x\\\\,x\\\\,y\\\\,z\\\\,z}}\\\\) for \\\\(f\\\\left( {x,y,z} \\\\right) = {z^3}{y^2}\\\\ln \\\\left( x \\\\right)\\\\)\n2. Find \\\\(\\\\displaystyle \\\\frac{{{\\\\partial ^3}f}}{{\\\\partial y\\\\partial {x^2}}}\\\\) for \\\\(f\\\\left( {x,y} \\\\right) = {{\\\\bf{e}}^{xy}}\\\\)\n\nShow All Solutions Hide All Solutions\n\na Find \\\\({f\\_{x\\\\,x\\\\,y\\\\,z\\\\,z}}\\\\) for \\\\(f\\\\left( {x,y,z} \\\\right) = {z^3}{y^2}\\\\ln \\\\left( x \\\\right)\\\\) Show Solution\n\nIn this case remember that we differentiate from left to right. Here are the derivatives for this part.\n\n\\\\\\[{f\\_x} = \\\\frac{{{z^3}{y^2}}}{x}\\\\\\] \\\\\\[{f\\_{xx}} = - \\\\frac{{{z^3}{y^2}}}{{{x^2}}}\\\\\\] \\\\\\[{f\\_{xxy}} = - \\\\frac{{2{z^3}y}}{{{x^2}}}\\\\\\] \\\\\\[{f\\_{xxyz}} = - \\\\frac{{6{z^2}y}}{{{x^2}}}\\\\\\] \\\\\\[{f\\_{xxyzz}} = - \\\\frac{{12zy}}{{{x^2}}}\\\\\\]\n\nb Find \\\\(\\\\displaystyle \\\\frac{{{\\\\partial ^3}f}}{{\\\\partial y\\\\partial {x^2}}}\\\\) for \\\\(f\\\\left( {x,y} \\\\right) = {{\\\\bf{e}}^{xy}}\\\\) Show Solution\n\nHere we differentiate from right to left. Here are the derivatives for this function.\n\n\\\\\\[\\\\frac{{\\\\partial f}}{{\\\\partial x}} = y{{\\\\bf{e}}^{xy}}\\\\\\] \\\\\\[\\\\frac{{{\\\\partial ^2}f}}{{\\\\partial {x^2}}} = {y^2}{{\\\\bf{e}}^{xy}}\\\\\\] \\\\\\[\\\\frac{{{\\\\partial ^3}f}}{{\\\\partial y\\\\partial {x^2}}} = 2y{{\\\\bf{e}}^{xy}} + x{y^2}{{\\\\bf{e}}^{xy}}\\\\\\]\n\n\\[[Contact Me](https:\/\/tutorial.math.lamar.edu\/Contact.aspx)\\] \\[[Privacy Statement](https:\/\/tutorial.math.lamar.edu\/Privacy.aspx)\\] \\[[Site Help & FAQ](https:\/\/tutorial.math.lamar.edu\/Help.aspx)\\] \\[[Terms of Use](https:\/\/tutorial.math.lamar.edu\/Terms.aspx)\\]\n\n© 2003 - 2026 Paul Dawkins\n\nPage Last Modified : 11\/16\/2022","attrs_readable_markdown":"Show Mobile Notice  Show All Notes Hide All Notes\n\nMobile Notice\n\nYou appear to be on a device with a \"narrow\" screen width (*i.e.* you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll\/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.\n### Section 13.4 : Higher Order Partial Derivatives\nJust as we had higher order derivatives with functions of one variable we will also have higher order derivatives of functions of more than one variable. However, this time we will have more options since we do have more than one variable.\n\nConsider the case of a function of two variables, \\\\(f\\\\left( {x,y} \\\\right)\\\\) since both of the first order partial derivatives are also functions of \\\\(x\\\\) and \\\\(y\\\\) we could in turn differentiate each with respect to \\\\(x\\\\) or \\\\(y\\\\). This means that for the case of a function of two variables there will be a total of four possible second order derivatives. Here they are and the notations that we’ll use to denote them.\n\n\\\\\\[\\\\begin{align\\*}{\\\\left( {{f\\_x}} \\\\right)\\_x} & = {f\\_{x\\\\,x}} = \\\\frac{\\\\partial }{{\\\\partial x}}\\\\left( {\\\\frac{{\\\\partial f}}{{\\\\partial x}}} \\\\right) = \\\\frac{{{\\\\partial ^2}f}}{{\\\\partial {x^2}}}\\\\\\\\ {\\\\left( {{f\\_x}} \\\\right)\\_y} & = {f\\_{x\\\\,y}} = \\\\frac{\\\\partial }{{\\\\partial y}}\\\\left( {\\\\frac{{\\\\partial f}}{{\\\\partial x}}} \\\\right) = \\\\frac{{{\\\\partial ^2}f}}{{\\\\partial y\\\\partial x}}\\\\\\\\ {\\\\left( {{f\\_y}} \\\\right)\\_x} & = {f\\_{y\\\\,x}} = \\\\frac{\\\\partial }{{\\\\partial x}}\\\\left( {\\\\frac{{\\\\partial f}}{{\\\\partial y}}} \\\\right) = \\\\frac{{{\\\\partial ^2}f}}{{\\\\partial x\\\\partial y}}\\\\\\\\ {\\\\left( {{f\\_y}} \\\\right)\\_y} & = {f\\_{y\\\\,y}} = \\\\frac{\\\\partial }{{\\\\partial y}}\\\\left( {\\\\frac{{\\\\partial f}}{{\\\\partial y}}} \\\\right) = \\\\frac{{{\\\\partial ^2}f}}{{\\\\partial {y^2}}}\\\\end{align\\*}\\\\\\]\n\nThe second and third second order partial derivatives are often called mixed partial derivatives since we are taking derivatives with respect to more than one variable. Note as well that the order that we take the derivatives in is given by the notation for each these. If we are using the subscripting notation, *e.g.* \\\\({f\\_{x\\\\,y}}\\\\), then we will differentiate from left to right. In other words, in this case, we will differentiate first with respect to \\\\(x\\\\) and then with respect to \\\\(y\\\\). With the fractional notation, *e.g.* \\\\(\\\\frac{{{\\\\partial ^2}f}}{{\\\\partial y\\\\partial x}}\\\\), it is the opposite. In these cases we differentiate moving along the denominator from right to left. So, again, in this case we differentiate with respect to \\\\(x\\\\) first and then \\\\(y\\\\).\n\nLet’s take a quick look at an example.\n\nExample 1 Find all the second order derivatives for \\\\(f\\\\left( {x,y} \\\\right) = \\\\cos \\\\left( {2x} \\\\right) - {x^2}{{\\\\bf{e}}^{5y}} + 3{y^2}\\\\).\n\nShow Solution\n\nWe’ll first need the first order derivatives so here they are.\n\n\\\\\\[\\\\begin{align\\*}{f\\_x}\\\\left( {x,y} \\\\right) & = - 2\\\\sin \\\\left( {2x} \\\\right) - 2x{{\\\\bf{e}}^{5y}}\\\\\\\\ {f\\_y}\\\\left( {x,y} \\\\right) & = - 5{x^2}{{\\\\bf{e}}^{5y}} + 6y\\\\end{align\\*}\\\\\\]\n\nNow, let’s get the second order derivatives.\n\n\\\\\\[\\\\begin{align\\*}{f\\_{xx}} & = - 4\\\\cos \\\\left( {2x} \\\\right) - 2{{\\\\bf{e}}^{5y}}\\\\\\\\ {f\\_{xy}} & = - 10x{{\\\\bf{e}}^{5y}}\\\\\\\\ {f\\_{yx}} & = - 10x{{\\\\bf{e}}^{5y}}\\\\\\\\ {f\\_{yy}} & = - 25{x^2}{{\\\\bf{e}}^{5y}} + 6\\\\end{align\\*}\\\\\\]\n\nNotice that we dropped the \\\\(\\\\left( {x,y} \\\\right)\\\\) from the derivatives. This is fairly standard and we will be doing it most of the time from this point on. We will also be dropping it for the first order derivatives in most cases.\n\nNow let’s also notice that, in this case, \\\\({f\\_{xy}} = {f\\_{yx}}\\\\). This is not by coincidence. If the function is “nice enough” this will always be the case. So, what’s “nice enough”? The following theorem tells us.\n\n#### Clairaut’s Theorem\nSuppose that \\\\(f\\\\) is defined on a disk \\\\(D\\\\) that contains the point \\\\(\\\\left( {a,b} \\\\right)\\\\). If the functions \\\\({f\\_{xy}}\\\\) and \\\\({f\\_{yx}}\\\\) are continuous on this disk then,\n\n\\\\\\[{f\\_{xy}}\\\\left( {a,b} \\\\right) = {f\\_{yx}}\\\\left( {a,b} \\\\right)\\\\\\]\n\nNow, do not get too excited about the disk business and the fact that we gave the theorem for a specific point. In pretty much every example in this class if the two mixed second order partial derivatives are continuous then they will be equal.\n\nExample 2 Verify Clairaut’s Theorem for \\\\(f\\\\left( {x,y} \\\\right) = x{{\\\\bf{e}}^{ - {x^2}{y^2}}}\\\\).\n\nShow Solution\n\nWe’ll first need the two first order derivatives.\n\n\\\\\\[\\\\begin{align\\*}{f\\_x}\\\\left( {x,y} \\\\right) & = {{\\\\bf{e}}^{ - {x^2}{y^2}}} - 2{x^2}{y^2}{{\\\\bf{e}}^{ - {x^2}{y^2}}}\\\\\\\\ {f\\_y}\\\\left( {x,y} \\\\right) & = - 2y{x^3}{{\\\\bf{e}}^{ - {x^2}{y^2}}}\\\\end{align\\*}\\\\\\]\n\nNow, compute the two mixed second order partial derivatives.\n\n\\\\\\[\\\\begin{align\\*}{f\\_{xy}}\\\\left( {x,y} \\\\right) & = - 2y{x^2}{{\\\\bf{e}}^{ - {x^2}{y^2}}} - 4{x^2}y{{\\\\bf{e}}^{ - {x^2}{y^2}}} + 4{x^4}{y^3}{{\\\\bf{e}}^{ - {x^2}{y^2}}} = - 6{x^2}y{{\\\\bf{e}}^{ - {x^2}{y^2}}} + 4{x^4}{y^3}{{\\\\bf{e}}^{ - {x^2}{y^2}}}\\\\\\\\ {f\\_{yx}}\\\\left( {x,y} \\\\right) & = - 6y{x^2}{{\\\\bf{e}}^{ - {x^2}{y^2}}} + 4{y^3}{x^4}{{\\\\bf{e}}^{ - {x^2}{y^2}}}\\\\end{align\\*}\\\\\\]\n\nSure enough they are the same.\n\nSo far we have only looked at second order derivatives. There are, of course, higher order derivatives as well. Here are a couple of the third order partial derivatives of function of two variables.\n\n\\\\\\[\\\\begin{align\\*}{f\\_{x\\\\,y\\\\,x}} & = {\\\\left( {{f\\_{xy}}} \\\\right)\\_x} = \\\\frac{\\\\partial }{{\\\\partial x}}\\\\left( {\\\\frac{{{\\\\partial ^2}f}}{{\\\\partial y\\\\partial x}}} \\\\right) = \\\\frac{{{\\\\partial ^3}f}}{{\\\\partial x\\\\partial y\\\\partial x}}\\\\\\\\ {f\\_{y\\\\,x\\\\,x}} & = {\\\\left( {{f\\_{yx}}} \\\\right)\\_x} = \\\\frac{\\\\partial }{{\\\\partial x}}\\\\left( {\\\\frac{{{\\\\partial ^2}f}}{{\\\\partial x\\\\partial y}}} \\\\right) = \\\\frac{{{\\\\partial ^3}f}}{{\\\\partial {x^2}\\\\partial y}}\\\\end{align\\*}\\\\\\]\n\nNotice as well that for both of these we differentiate once with respect to \\\\(y\\\\) and twice with respect to \\\\(x\\\\). There is also another third order partial derivative in which we can do this, \\\\({f\\_{x\\\\,x\\\\,y}}\\\\). There is an extension to Clairaut’s Theorem that says if all three of these are continuous then they should all be equal,\n\n\\\\\\[{f\\_{x\\\\,x\\\\,y}} = {f\\_{x\\\\,y\\\\,x}} = {f\\_{y\\\\,x\\\\,x}}\\\\\\]\n\nTo this point we’ve only looked at functions of two variables, but everything that we’ve done to this point will work regardless of the number of variables that we’ve got in the function and there are natural extensions to Clairaut’s theorem to all of these cases as well. For instance,\n\n\\\\\\[{f\\_{x\\\\,z}}\\\\left( {x,y,z} \\\\right) = {f\\_{z\\\\,x}}\\\\left( {x,y,z} \\\\right)\\\\\\]\n\nprovided both of the derivatives are continuous.\n\nIn general, we can extend Clairaut’s theorem to any function and mixed partial derivatives. The only requirement is that in each derivative we differentiate with respect to each variable the same number of times. In other words, provided we meet the continuity condition, the following will be equal\n\n\\\\\\[{f\\_{s\\\\,s\\\\,r\\\\,t\\\\,s\\\\,r\\\\,r}} = {f\\_{t\\\\,r\\\\,s\\\\,r\\\\,s\\\\,s\\\\,r}}\\\\\\]\n\nbecause in each case we differentiate with respect to \\\\(t\\\\) once, \\\\(s\\\\) three times and \\\\(r\\\\) three times.\n\nLet’s do a couple of examples with higher (well higher order than two anyway) order derivatives and functions of more than two variables.\n\nExample 3 Find the indicated derivative for each of the following functions.\n\n1. Find \\\\({f\\_{x\\\\,x\\\\,y\\\\,z\\\\,z}}\\\\) for \\\\(f\\\\left( {x,y,z} \\\\right) = {z^3}{y^2}\\\\ln \\\\left( x \\\\right)\\\\)\n2. Find \\\\(\\\\displaystyle \\\\frac{{{\\\\partial ^3}f}}{{\\\\partial y\\\\partial {x^2}}}\\\\) for \\\\(f\\\\left( {x,y} \\\\right) = {{\\\\bf{e}}^{xy}}\\\\)\n\nShow All Solutions Hide All Solutions\n\na Find \\\\({f\\_{x\\\\,x\\\\,y\\\\,z\\\\,z}}\\\\) for \\\\(f\\\\left( {x,y,z} \\\\right) = {z^3}{y^2}\\\\ln \\\\left( x \\\\right)\\\\) Show Solution\n\nIn this case remember that we differentiate from left to right. Here are the derivatives for this part.\n\n\\\\\\[{f\\_x} = \\\\frac{{{z^3}{y^2}}}{x}\\\\\\] \\\\\\[{f\\_{xx}} = - \\\\frac{{{z^3}{y^2}}}{{{x^2}}}\\\\\\] \\\\\\[{f\\_{xxy}} = - \\\\frac{{2{z^3}y}}{{{x^2}}}\\\\\\] \\\\\\[{f\\_{xxyz}} = - \\\\frac{{6{z^2}y}}{{{x^2}}}\\\\\\] \\\\\\[{f\\_{xxyzz}} = - \\\\frac{{12zy}}{{{x^2}}}\\\\\\]\n\nb Find \\\\(\\\\displaystyle \\\\frac{{{\\\\partial ^3}f}}{{\\\\partial y\\\\partial {x^2}}}\\\\) for \\\\(f\\\\left( {x,y} \\\\right) = {{\\\\bf{e}}^{xy}}\\\\) Show Solution\n\nHere we differentiate from right to left. Here are the derivatives for this function.\n\n\\\\\\[\\\\frac{{\\\\partial f}}{{\\\\partial x}} = y{{\\\\bf{e}}^{xy}}\\\\\\] \\\\\\[\\\\frac{{{\\\\partial ^2}f}}{{\\\\partial {x^2}}} = {y^2}{{\\\\bf{e}}^{xy}}\\\\\\] \\\\\\[\\\\frac{{{\\\\partial ^3}f}}{{\\\\partial y\\\\partial {x^2}}} = 2y{{\\\\bf{e}}^{xy}} + x{y^2}{{\\\\bf{e}}^{xy}}\\\\\\]","meta_canonical":null}

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Meta Description	In the section we will take a look at higher order partial derivatives. Unlike Calculus I however, we will have multiple second order derivatives, multiple third order derivatives, etc. because we are now working with functions of multiple variables. We will also discuss Clairaut’s Theorem to help with some of the work in finding higher order derivatives.
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Boilerpipe Text	Show Mobile Notice Show All Notes Hide All Notes Mobile Notice You appear to be on a device with a "narrow" screen width ( i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width. Section 13.4 : Higher Order Partial Derivatives Just as we had higher order derivatives with functions of one variable we will also have higher order derivatives of functions of more than one variable. However, this time we will have more options since we do have more than one variable. Consider the case of a function of two variables, \(f\left( {x,y} \right)\) since both of the first order partial derivatives are also functions of \(x\) and \(y\) we could in turn differentiate each with respect to \(x\) or \(y\). This means that for the case of a function of two variables there will be a total of four possible second order derivatives. Here they are and the notations that we’ll use to denote them. \[\begin{align}{\left( {{f_x}} \right)_x} & = {f_{x\,x}} = \frac{\partial }{{\partial x}}\left( {\frac{{\partial f}}{{\partial x}}} \right) = \frac{{{\partial ^2}f}}{{\partial {x^2}}}\\ {\left( {{f_x}} \right)_y} & = {f_{x\,y}} = \frac{\partial }{{\partial y}}\left( {\frac{{\partial f}}{{\partial x}}} \right) = \frac{{{\partial ^2}f}}{{\partial y\partial x}}\\ {\left( {{f_y}} \right)_x} & = {f_{y\,x}} = \frac{\partial }{{\partial x}}\left( {\frac{{\partial f}}{{\partial y}}} \right) = \frac{{{\partial ^2}f}}{{\partial x\partial y}}\\ {\left( {{f_y}} \right)_y} & = {f_{y\,y}} = \frac{\partial }{{\partial y}}\left( {\frac{{\partial f}}{{\partial y}}} \right) = \frac{{{\partial ^2}f}}{{\partial {y^2}}}\end{align}\] The second and third second order partial derivatives are often called mixed partial derivatives since we are taking derivatives with respect to more than one variable. Note as well that the order that we take the derivatives in is given by the notation for each these. If we are using the subscripting notation, e.g. \({f_{x\,y}}\), then we will differentiate from left to right. In other words, in this case, we will differentiate first with respect to \(x\) and then with respect to \(y\). With the fractional notation, e.g. \(\frac{{{\partial ^2}f}}{{\partial y\partial x}}\), it is the opposite. In these cases we differentiate moving along the denominator from right to left. So, again, in this case we differentiate with respect to \(x\) first and then \(y\). Let’s take a quick look at an example. Example 1 Find all the second order derivatives for \(f\left( {x,y} \right) = \cos \left( {2x} \right) - {x^2}{{\bf{e}}^{5y}} + 3{y^2}\). Show Solution We’ll first need the first order derivatives so here they are. \[\begin{align}{f_x}\left( {x,y} \right) & = - 2\sin \left( {2x} \right) - 2x{{\bf{e}}^{5y}}\\ {f_y}\left( {x,y} \right) & = - 5{x^2}{{\bf{e}}^{5y}} + 6y\end{align}\] Now, let’s get the second order derivatives. \[\begin{align}{f_{xx}} & = - 4\cos \left( {2x} \right) - 2{{\bf{e}}^{5y}}\\ {f_{xy}} & = - 10x{{\bf{e}}^{5y}}\\ {f_{yx}} & = - 10x{{\bf{e}}^{5y}}\\ {f_{yy}} & = - 25{x^2}{{\bf{e}}^{5y}} + 6\end{align}\] Notice that we dropped the \(\left( {x,y} \right)\) from the derivatives. This is fairly standard and we will be doing it most of the time from this point on. We will also be dropping it for the first order derivatives in most cases. Now let’s also notice that, in this case, \({f_{xy}} = {f_{yx}}\). This is not by coincidence. If the function is “nice enough” this will always be the case. So, what’s “nice enough”? The following theorem tells us. Clairaut’s Theorem Suppose that \(f\) is defined on a disk \(D\) that contains the point \(\left( {a,b} \right)\). If the functions \({f_{xy}}\) and \({f_{yx}}\) are continuous on this disk then, \[{f_{xy}}\left( {a,b} \right) = {f_{yx}}\left( {a,b} \right)\] Now, do not get too excited about the disk business and the fact that we gave the theorem for a specific point. In pretty much every example in this class if the two mixed second order partial derivatives are continuous then they will be equal. Example 2 Verify Clairaut’s Theorem for \(f\left( {x,y} \right) = x{{\bf{e}}^{ - {x^2}{y^2}}}\). Show Solution We’ll first need the two first order derivatives. \[\begin{align}{f_x}\left( {x,y} \right) & = {{\bf{e}}^{ - {x^2}{y^2}}} - 2{x^2}{y^2}{{\bf{e}}^{ - {x^2}{y^2}}}\\ {f_y}\left( {x,y} \right) & = - 2y{x^3}{{\bf{e}}^{ - {x^2}{y^2}}}\end{align}\] Now, compute the two mixed second order partial derivatives. \[\begin{align}{f_{xy}}\left( {x,y} \right) & = - 2y{x^2}{{\bf{e}}^{ - {x^2}{y^2}}} - 4{x^2}y{{\bf{e}}^{ - {x^2}{y^2}}} + 4{x^4}{y^3}{{\bf{e}}^{ - {x^2}{y^2}}} = - 6{x^2}y{{\bf{e}}^{ - {x^2}{y^2}}} + 4{x^4}{y^3}{{\bf{e}}^{ - {x^2}{y^2}}}\\ {f_{yx}}\left( {x,y} \right) & = - 6y{x^2}{{\bf{e}}^{ - {x^2}{y^2}}} + 4{y^3}{x^4}{{\bf{e}}^{ - {x^2}{y^2}}}\end{align}\] Sure enough they are the same. So far we have only looked at second order derivatives. There are, of course, higher order derivatives as well. Here are a couple of the third order partial derivatives of function of two variables. \[\begin{align}{f_{x\,y\,x}} & = {\left( {{f_{xy}}} \right)_x} = \frac{\partial }{{\partial x}}\left( {\frac{{{\partial ^2}f}}{{\partial y\partial x}}} \right) = \frac{{{\partial ^3}f}}{{\partial x\partial y\partial x}}\\ {f_{y\,x\,x}} & = {\left( {{f_{yx}}} \right)_x} = \frac{\partial }{{\partial x}}\left( {\frac{{{\partial ^2}f}}{{\partial x\partial y}}} \right) = \frac{{{\partial ^3}f}}{{\partial {x^2}\partial y}}\end{align}\] Notice as well that for both of these we differentiate once with respect to \(y\) and twice with respect to \(x\). There is also another third order partial derivative in which we can do this, \({f_{x\,x\,y}}\). There is an extension to Clairaut’s Theorem that says if all three of these are continuous then they should all be equal, \[{f_{x\,x\,y}} = {f_{x\,y\,x}} = {f_{y\,x\,x}}\] To this point we’ve only looked at functions of two variables, but everything that we’ve done to this point will work regardless of the number of variables that we’ve got in the function and there are natural extensions to Clairaut’s theorem to all of these cases as well. For instance, \[{f_{x\,z}}\left( {x,y,z} \right) = {f_{z\,x}}\left( {x,y,z} \right)\] provided both of the derivatives are continuous. In general, we can extend Clairaut’s theorem to any function and mixed partial derivatives. The only requirement is that in each derivative we differentiate with respect to each variable the same number of times. In other words, provided we meet the continuity condition, the following will be equal \[{f_{s\,s\,r\,t\,s\,r\,r}} = {f_{t\,r\,s\,r\,s\,s\,r}}\] because in each case we differentiate with respect to \(t\) once, \(s\) three times and \(r\) three times. Let’s do a couple of examples with higher (well higher order than two anyway) order derivatives and functions of more than two variables. Example 3 Find the indicated derivative for each of the following functions. Find \({f_{x\,x\,y\,z\,z}}\) for \(f\left( {x,y,z} \right) = {z^3}{y^2}\ln \left( x \right)\) Find \(\displaystyle \frac{{{\partial ^3}f}}{{\partial y\partial {x^2}}}\) for \(f\left( {x,y} \right) = {{\bf{e}}^{xy}}\) Show All Solutions Hide All Solutions a Find \({f_{x\,x\,y\,z\,z}}\) for \(f\left( {x,y,z} \right) = {z^3}{y^2}\ln \left( x \right)\) Show Solution In this case remember that we differentiate from left to right. Here are the derivatives for this part. \[{f_x} = \frac{{{z^3}{y^2}}}{x}\] \[{f_{xx}} = - \frac{{{z^3}{y^2}}}{{{x^2}}}\] \[{f_{xxy}} = - \frac{{2{z^3}y}}{{{x^2}}}\] \[{f_{xxyz}} = - \frac{{6{z^2}y}}{{{x^2}}}\] \[{f_{xxyzz}} = - \frac{{12zy}}{{{x^2}}}\] b Find \(\displaystyle \frac{{{\partial ^3}f}}{{\partial y\partial {x^2}}}\) for \(f\left( {x,y} \right) = {{\bf{e}}^{xy}}\) Show Solution Here we differentiate from right to left. Here are the derivatives for this function. \[\frac{{\partial f}}{{\partial x}} = y{{\bf{e}}^{xy}}\] \[\frac{{{\partial ^2}f}}{{\partial {x^2}}} = {y^2}{{\bf{e}}^{xy}}\] \[\frac{{{\partial ^3}f}}{{\partial y\partial {x^2}}} = 2y{{\bf{e}}^{xy}} + x{y^2}{{\bf{e}}^{xy}}\]
Markdown	[Paul's Online Notes](https://tutorial.math.lamar.edu/) Notes Quick Nav Download - Go To - Notes - [Practice Problems](https://tutorial.math.lamar.edu/Problems/CalcIII/HighOrderPartialDerivs.aspx "Go to Practice Problems for current topic.") - [Assignment Problems](https://tutorial.math.lamar.edu/ProblemsNS/CalcIII/HighOrderPartialDerivs.aspx "Go to Assignment Problems for current topic.") - Show/Hide - [Show all Solutions/Steps/etc.]() - [Hide all Solutions/Steps/etc.]() - Sections - [Interpretations of Partial Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/PartialDerivInterp.aspx "Previous Section : Interpretations of Partial Derivatives") - [Differentials](https://tutorial.math.lamar.edu/Classes/CalcIII/Differentials.aspx "Next Section : Differentials") - Chapters - [3-Dimensional Space](https://tutorial.math.lamar.edu/Classes/CalcIII/3DSpace.aspx "Previous Chapter : 3-Dimensional Space") - [Applications of Partial Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/PartialDerivAppsIntro.aspx "Next Chapter : Applications of Partial Derivatives") - Classes - [Algebra](https://tutorial.math.lamar.edu/Classes/Alg/Alg.aspx "Go To Algebra Notes") - [Calculus I](https://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx "Go To Calculus I Notes") - [Calculus II](https://tutorial.math.lamar.edu/Classes/CalcII/CalcII.aspx "Go To Calculus II Notes") - Calculus III - [Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/DE.aspx "Go To Differential Equations Notes") - Extras - [Algebra & Trig Review](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/AlgebraTrig.aspx "Go To Algebra & Trig Review") - [Common Math Errors](https://tutorial.math.lamar.edu/Extras/CommonErrors/CommonMathErrors.aspx "Go To Common Math Errors") - [Complex Number Primer](https://tutorial.math.lamar.edu/Extras/ComplexPrimer/ComplexNumbers.aspx "Go To Complex Numbers Primer") - [How To Study Math](https://tutorial.math.lamar.edu/Extras/StudyMath/HowToStudyMath.aspx "Go To How To Study Math") - [Cheat Sheets & Tables](https://tutorial.math.lamar.edu/Extras/CheatSheets_Tables.aspx "Go To List of Cheat Sheets and Tables") - Misc - [Contact Me](https://tutorial.math.lamar.edu/contact.aspx "Contact Me!") - [MathJax Help and Configuration](https://tutorial.math.lamar.edu/mathjax.aspx "Info on MathJax and MathJax Configuration Menu") - Notes Downloads - [Complete Book](https://tutorial.math.lamar.edu/GetFile.aspx?file=B,20,N) - Practice Problems Downloads - [Complete Book - Problems Only](https://tutorial.math.lamar.edu/GetFile.aspx?file=B,20,P) - [Complete Book - Solutions](https://tutorial.math.lamar.edu/GetFile.aspx?file=B,20,S) - Assignment Problems Downloads - [Complete Book](https://tutorial.math.lamar.edu/GetFile.aspx?file=B,20,A) - Other Items - [Get URL's for Download Items](https://tutorial.math.lamar.edu/DownloadURLs.aspx?bi=20) - [Print Page in Current Form (Default)]() - [Show all Solutions/Steps and Print Page]() - [Hide all Solutions/Steps and Print Page]() - [Home](https://tutorial.math.lamar.edu/) - Classes - [Algebra](https://tutorial.math.lamar.edu/Classes/Alg/Alg.aspx) - [1\. Preliminaries](https://tutorial.math.lamar.edu/Classes/Alg/Preliminaries.aspx) - [1\.1 Integer Exponents](https://tutorial.math.lamar.edu/Classes/Alg/IntegerExponents.aspx) - [1\.2 Rational Exponents](https://tutorial.math.lamar.edu/Classes/Alg/RationalExponents.aspx) - [1\.3 Radicals](https://tutorial.math.lamar.edu/Classes/Alg/Radicals.aspx) - [1\.4 Polynomials](https://tutorial.math.lamar.edu/Classes/Alg/Polynomials.aspx) - [1\.5 Factoring Polynomials](https://tutorial.math.lamar.edu/Classes/Alg/Factoring.aspx) - [1\.6 Rational Expressions](https://tutorial.math.lamar.edu/Classes/Alg/RationalExpressions.aspx) - [1\.7 Complex Numbers](https://tutorial.math.lamar.edu/Classes/Alg/ComplexNumbers.aspx) - [2\. Solving Equations and Inequalities](https://tutorial.math.lamar.edu/Classes/Alg/Solving.aspx) - [2\.1 Solutions and Solution Sets](https://tutorial.math.lamar.edu/Classes/Alg/SolutionSets.aspx) - [2\.2 Linear Equations](https://tutorial.math.lamar.edu/Classes/Alg/SolveLinearEqns.aspx) - [2\.3 Applications of Linear Equations](https://tutorial.math.lamar.edu/Classes/Alg/LinearApps.aspx) - [2\.4 Equations With More Than One Variable](https://tutorial.math.lamar.edu/Classes/Alg/SolveMultiVariable.aspx) - [2\.5 Quadratic Equations - Part I](https://tutorial.math.lamar.edu/Classes/Alg/SolveQuadraticEqnsI.aspx) - [2\.6 Quadratic Equations - Part II](https://tutorial.math.lamar.edu/Classes/Alg/SolveQuadraticEqnsII.aspx) - [2\.7 Quadratic Equations : A Summary](https://tutorial.math.lamar.edu/Classes/Alg/SolveQuadraticEqnSummary.aspx) - [2\.8 Applications of Quadratic Equations](https://tutorial.math.lamar.edu/Classes/Alg/QuadraticApps.aspx) - [2\.9 Equations Reducible to Quadratic in Form](https://tutorial.math.lamar.edu/Classes/Alg/ReducibleToQuadratic.aspx) - [2\.10 Equations with Radicals](https://tutorial.math.lamar.edu/Classes/Alg/SolveRadicalEqns.aspx) - [2\.11 Linear Inequalities](https://tutorial.math.lamar.edu/Classes/Alg/SolveLinearInequalities.aspx) - [2\.12 Polynomial Inequalities](https://tutorial.math.lamar.edu/Classes/Alg/SolvePolyInequalities.aspx) - [2\.13 Rational Inequalities](https://tutorial.math.lamar.edu/Classes/Alg/SolveRationalInequalities.aspx) - [2\.14 Absolute Value Equations](https://tutorial.math.lamar.edu/Classes/Alg/SolveAbsValueEqns.aspx) - [2\.15 Absolute Value Inequalities](https://tutorial.math.lamar.edu/Classes/Alg/SolveAbsValueIneq.aspx) - [3\. Graphing and Functions](https://tutorial.math.lamar.edu/Classes/Alg/Graphing_Functions.aspx) - [3\.1 Graphing](https://tutorial.math.lamar.edu/Classes/Alg/Graphing.aspx) - [3\.2 Lines](https://tutorial.math.lamar.edu/Classes/Alg/Lines.aspx) - [3\.3 Circles](https://tutorial.math.lamar.edu/Classes/Alg/Circles.aspx) - [3\.4 The Definition of a Function](https://tutorial.math.lamar.edu/Classes/Alg/FunctionDefn.aspx) - [3\.5 Graphing Functions](https://tutorial.math.lamar.edu/Classes/Alg/GraphFunctions.aspx) - [3\.6 Combining Functions](https://tutorial.math.lamar.edu/Classes/Alg/CombineFunctions.aspx) - [3\.7 Inverse Functions](https://tutorial.math.lamar.edu/Classes/Alg/InverseFunctions.aspx) - [4\. Common Graphs](https://tutorial.math.lamar.edu/Classes/Alg/CommonGraphs.aspx) - [4\.1 Lines, Circles and Piecewise Functions](https://tutorial.math.lamar.edu/Classes/Alg/Lines_Circles_PWF.aspx) - [4\.2 Parabolas](https://tutorial.math.lamar.edu/Classes/Alg/Parabolas.aspx) - [4\.3 Ellipses](https://tutorial.math.lamar.edu/Classes/Alg/Ellipses.aspx) - [4\.4 Hyperbolas](https://tutorial.math.lamar.edu/Classes/Alg/Hyperbolas.aspx) - [4\.5 Miscellaneous Functions](https://tutorial.math.lamar.edu/Classes/Alg/MiscFunctions.aspx) - [4\.6 Transformations](https://tutorial.math.lamar.edu/Classes/Alg/Transformations.aspx) - [4\.7 Symmetry](https://tutorial.math.lamar.edu/Classes/Alg/Symmetry.aspx) - [4\.8 Rational Functions](https://tutorial.math.lamar.edu/Classes/Alg/GraphRationalFcns.aspx) - [5\. Polynomial Functions](https://tutorial.math.lamar.edu/Classes/Alg/PolynomialFunctions.aspx) - [5\.1 Dividing Polynomials](https://tutorial.math.lamar.edu/Classes/Alg/DividingPolynomials.aspx) - [5\.2 Zeroes/Roots of Polynomials](https://tutorial.math.lamar.edu/Classes/Alg/ZeroesOfPolynomials.aspx) - [5\.3 Graphing Polynomials](https://tutorial.math.lamar.edu/Classes/Alg/GraphingPolynomials.aspx) - [5\.4 Finding Zeroes of Polynomials](https://tutorial.math.lamar.edu/Classes/Alg/FindingZeroesOfPolynomials.aspx) - [5\.5 Partial Fractions](https://tutorial.math.lamar.edu/Classes/Alg/PartialFractions.aspx) - [6\. Exponential and Logarithm Functions](https://tutorial.math.lamar.edu/Classes/Alg/ExpAndLog.aspx) - [6\.1 Exponential Functions](https://tutorial.math.lamar.edu/Classes/Alg/ExpFunctions.aspx) - [6\.2 Logarithm Functions](https://tutorial.math.lamar.edu/Classes/Alg/LogFunctions.aspx) - [6\.3 Solving Exponential Equations](https://tutorial.math.lamar.edu/Classes/Alg/SolveExpEqns.aspx) - [6\.4 Solving Logarithm Equations](https://tutorial.math.lamar.edu/Classes/Alg/SolveLogEqns.aspx) - [6\.5 Applications](https://tutorial.math.lamar.edu/Classes/Alg/ExpLogApplications.aspx) - [7\. Systems of Equations](https://tutorial.math.lamar.edu/Classes/Alg/Systems.aspx) - [7\.1 Linear Systems with Two Variables](https://tutorial.math.lamar.edu/Classes/Alg/SystemsTwoVrble.aspx) - [7\.2 Linear Systems with Three Variables](https://tutorial.math.lamar.edu/Classes/Alg/SystemsThreeVrble.aspx) - [7\.3 Augmented Matrices](https://tutorial.math.lamar.edu/Classes/Alg/AugmentedMatrix.aspx) - [7\.4 More on the Augmented Matrix](https://tutorial.math.lamar.edu/Classes/Alg/AugmentedMatrixII.aspx) - [7\.5 Nonlinear Systems](https://tutorial.math.lamar.edu/Classes/Alg/NonlinearSystems.aspx) - [Calculus I](https://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx) - [1\. Review](https://tutorial.math.lamar.edu/Classes/CalcI/ReviewIntro.aspx) - [1\.1 Functions](https://tutorial.math.lamar.edu/Classes/CalcI/Functions.aspx) - [1\.2 Inverse Functions](https://tutorial.math.lamar.edu/Classes/CalcI/InverseFunctions.aspx) - [1\.3 Trig Functions](https://tutorial.math.lamar.edu/Classes/CalcI/TrigFcns.aspx) - [1\.4 Solving Trig Equations](https://tutorial.math.lamar.edu/Classes/CalcI/TrigEquations.aspx) - [1\.5 Trig Equations with Calculators, Part I](https://tutorial.math.lamar.edu/Classes/CalcI/TrigEquations_CalcI.aspx) - [1\.6 Trig Equations with Calculators, Part II](https://tutorial.math.lamar.edu/Classes/CalcI/TrigEquations_CalcII.aspx) - [1\.7 Exponential Functions](https://tutorial.math.lamar.edu/Classes/CalcI/ExpFunctions.aspx) - [1\.8 Logarithm Functions](https://tutorial.math.lamar.edu/Classes/CalcI/LogFcns.aspx) - [1\.9 Exponential and Logarithm Equations](https://tutorial.math.lamar.edu/Classes/CalcI/ExpLogEqns.aspx) - [1\.10 Common Graphs](https://tutorial.math.lamar.edu/Classes/CalcI/CommonGraphs.aspx) - [2\. Limits](https://tutorial.math.lamar.edu/Classes/CalcI/limitsIntro.aspx) - [2\.1 Tangent Lines and Rates of Change](https://tutorial.math.lamar.edu/Classes/CalcI/Tangents_Rates.aspx) - [2\.2 The Limit](https://tutorial.math.lamar.edu/Classes/CalcI/TheLimit.aspx) - [2\.3 One-Sided Limits](https://tutorial.math.lamar.edu/Classes/CalcI/OneSidedLimits.aspx) - [2\.4 Limit Properties](https://tutorial.math.lamar.edu/Classes/CalcI/LimitsProperties.aspx) - [2\.5 Computing Limits](https://tutorial.math.lamar.edu/Classes/CalcI/ComputingLimits.aspx) - [2\.6 Infinite Limits](https://tutorial.math.lamar.edu/Classes/CalcI/InfiniteLimits.aspx) - [2\.7 Limits At Infinity, Part I](https://tutorial.math.lamar.edu/Classes/CalcI/LimitsAtInfinityI.aspx) - [2\.8 Limits At Infinity, Part II](https://tutorial.math.lamar.edu/Classes/CalcI/LimitsAtInfinityII.aspx) - [2\.9 Continuity](https://tutorial.math.lamar.edu/Classes/CalcI/Continuity.aspx) - [2\.10 The Definition of the Limit](https://tutorial.math.lamar.edu/Classes/CalcI/DefnOfLimit.aspx) - [3\. Derivatives](https://tutorial.math.lamar.edu/Classes/CalcI/DerivativeIntro.aspx) - [3\.1 The Definition of the Derivative](https://tutorial.math.lamar.edu/Classes/CalcI/DefnOfDerivative.aspx) - [3\.2 Interpretation of the Derivative](https://tutorial.math.lamar.edu/Classes/CalcI/DerivativeInterp.aspx) - [3\.3 Differentiation Formulas](https://tutorial.math.lamar.edu/Classes/CalcI/DiffFormulas.aspx) - [3\.4 Product and Quotient Rule](https://tutorial.math.lamar.edu/Classes/CalcI/ProductQuotientRule.aspx) - [3\.5 Derivatives of Trig Functions](https://tutorial.math.lamar.edu/Classes/CalcI/DiffTrigFcns.aspx) - [3\.6 Derivatives of Exponential and Logarithm Functions](https://tutorial.math.lamar.edu/Classes/CalcI/DiffExpLogFcns.aspx) - [3\.7 Derivatives of Inverse Trig Functions](https://tutorial.math.lamar.edu/Classes/CalcI/DiffInvTrigFcns.aspx) - [3\.8 Derivatives of Hyperbolic Functions](https://tutorial.math.lamar.edu/Classes/CalcI/DiffHyperFcns.aspx) - [3\.9 Chain Rule](https://tutorial.math.lamar.edu/Classes/CalcI/ChainRule.aspx) - [3\.10 Implicit Differentiation](https://tutorial.math.lamar.edu/Classes/CalcI/ImplicitDIff.aspx) - [3\.11 Related Rates](https://tutorial.math.lamar.edu/Classes/CalcI/RelatedRates.aspx) - [3\.12 Higher Order Derivatives](https://tutorial.math.lamar.edu/Classes/CalcI/HigherOrderDerivatives.aspx) - [3\.13 Logarithmic Differentiation](https://tutorial.math.lamar.edu/Classes/CalcI/LogDiff.aspx) - [4\. Applications of Derivatives](https://tutorial.math.lamar.edu/Classes/CalcI/DerivAppsIntro.aspx) - [4\.1 Rates of Change](https://tutorial.math.lamar.edu/Classes/CalcI/RateOfChange.aspx) - [4\.2 Critical Points](https://tutorial.math.lamar.edu/Classes/CalcI/CriticalPoints.aspx) - [4\.3 Minimum and Maximum Values](https://tutorial.math.lamar.edu/Classes/CalcI/MinMaxValues.aspx) - [4\.4 Finding Absolute Extrema](https://tutorial.math.lamar.edu/Classes/CalcI/AbsExtrema.aspx) - [4\.5 The Shape of a Graph, Part I](https://tutorial.math.lamar.edu/Classes/CalcI/ShapeofGraphPtI.aspx) - [4\.6 The Shape of a Graph, Part II](https://tutorial.math.lamar.edu/Classes/CalcI/ShapeofGraphPtII.aspx) - [4\.7 The Mean Value Theorem](https://tutorial.math.lamar.edu/Classes/CalcI/MeanValueTheorem.aspx) - [4\.8 Optimization](https://tutorial.math.lamar.edu/Classes/CalcI/Optimization.aspx) - [4\.9 More Optimization Problems](https://tutorial.math.lamar.edu/Classes/CalcI/MoreOptimization.aspx) - [4\.10 L'Hospital's Rule and Indeterminate Forms](https://tutorial.math.lamar.edu/Classes/CalcI/LHospitalsRule.aspx) - [4\.11 Linear Approximations](https://tutorial.math.lamar.edu/Classes/CalcI/LinearApproximations.aspx) - [4\.12 Differentials](https://tutorial.math.lamar.edu/Classes/CalcI/Differentials.aspx) - [4\.13 Newton's Method](https://tutorial.math.lamar.edu/Classes/CalcI/NewtonsMethod.aspx) - [4\.14 Business Applications](https://tutorial.math.lamar.edu/Classes/CalcI/BusinessApps.aspx) - [5\. Integrals](https://tutorial.math.lamar.edu/Classes/CalcI/IntegralsIntro.aspx) - [5\.1 Indefinite Integrals](https://tutorial.math.lamar.edu/Classes/CalcI/IndefiniteIntegrals.aspx) - [5\.2 Computing Indefinite Integrals](https://tutorial.math.lamar.edu/Classes/CalcI/ComputingIndefiniteIntegrals.aspx) - [5\.3 Substitution Rule for Indefinite Integrals](https://tutorial.math.lamar.edu/Classes/CalcI/SubstitutionRuleIndefinite.aspx) - [5\.4 More Substitution Rule](https://tutorial.math.lamar.edu/Classes/CalcI/SubstitutionRuleIndefinitePtII.aspx) - [5\.5 Area Problem](https://tutorial.math.lamar.edu/Classes/CalcI/AreaProblem.aspx) - [5\.6 Definition of the Definite Integral](https://tutorial.math.lamar.edu/Classes/CalcI/DefnOfDefiniteIntegral.aspx) - [5\.7 Computing Definite Integrals](https://tutorial.math.lamar.edu/Classes/CalcI/ComputingDefiniteIntegrals.aspx) - [5\.8 Substitution Rule for Definite Integrals](https://tutorial.math.lamar.edu/Classes/CalcI/SubstitutionRuleDefinite.aspx) - [6\. Applications of Integrals](https://tutorial.math.lamar.edu/Classes/CalcI/IntAppsIntro.aspx) - [6\.1 Average Function Value](https://tutorial.math.lamar.edu/Classes/CalcI/AvgFcnValue.aspx) - [6\.2 Area Between Curves](https://tutorial.math.lamar.edu/Classes/CalcI/AreaBetweenCurves.aspx) - [6\.3 Volumes of Solids of Revolution / Method of Rings](https://tutorial.math.lamar.edu/Classes/CalcI/VolumeWithRings.aspx) - [6\.4 Volumes of Solids of Revolution/Method of Cylinders](https://tutorial.math.lamar.edu/Classes/CalcI/VolumeWithCylinder.aspx) - [6\.5 More Volume Problems](https://tutorial.math.lamar.edu/Classes/CalcI/MoreVolume.aspx) - [6\.6 Work](https://tutorial.math.lamar.edu/Classes/CalcI/Work.aspx) - [Appendix A. Extras](https://tutorial.math.lamar.edu/Classes/CalcI/ExtrasIntro.aspx) - [A.1 Proof of Various Limit Properties](https://tutorial.math.lamar.edu/Classes/CalcI/LimitProofs.aspx) - [A.2 Proof of Various Derivative Properties](https://tutorial.math.lamar.edu/Classes/CalcI/DerivativeProofs.aspx) - [A.3 Proof of Trig Limits](https://tutorial.math.lamar.edu/Classes/CalcI/ProofTrigDeriv.aspx) - [A.4 Proofs of Derivative Applications Facts](https://tutorial.math.lamar.edu/Classes/CalcI/DerivativeAppsProofs.aspx) - [A.5 Proof of Various Integral Properties](https://tutorial.math.lamar.edu/Classes/CalcI/ProofIntProp.aspx) - [A.6 Area and Volume Formulas](https://tutorial.math.lamar.edu/Classes/CalcI/Area_Volume_Formulas.aspx) - [A.7 Types of Infinity](https://tutorial.math.lamar.edu/Classes/CalcI/TypesOfInfinity.aspx) - [A.8 Summation Notation](https://tutorial.math.lamar.edu/Classes/CalcI/SummationNotation.aspx) - [A.9 Constant of Integration](https://tutorial.math.lamar.edu/Classes/CalcI/ConstantofIntegration.aspx) - [Calculus II](https://tutorial.math.lamar.edu/Classes/CalcII/CalcII.aspx) - [7\. Integration Techniques](https://tutorial.math.lamar.edu/Classes/CalcII/IntTechIntro.aspx) - [7\.1 Integration by Parts](https://tutorial.math.lamar.edu/Classes/CalcII/IntegrationByParts.aspx) - [7\.2 Integrals Involving Trig Functions](https://tutorial.math.lamar.edu/Classes/CalcII/IntegralsWithTrig.aspx) - [7\.3 Trig Substitutions](https://tutorial.math.lamar.edu/Classes/CalcII/TrigSubstitutions.aspx) - [7\.4 Partial Fractions](https://tutorial.math.lamar.edu/Classes/CalcII/PartialFractions.aspx) - [7\.5 Integrals Involving Roots](https://tutorial.math.lamar.edu/Classes/CalcII/IntegralsWithRoots.aspx) - [7\.6 Integrals Involving Quadratics](https://tutorial.math.lamar.edu/Classes/CalcII/IntegralsWithQuadratics.aspx) - [7\.7 Integration Strategy](https://tutorial.math.lamar.edu/Classes/CalcII/IntegrationStrategy.aspx) - [7\.8 Improper Integrals](https://tutorial.math.lamar.edu/Classes/CalcII/ImproperIntegrals.aspx) - [7\.9 Comparison Test for Improper Integrals](https://tutorial.math.lamar.edu/Classes/CalcII/ImproperIntegralsCompTest.aspx) - [7\.10 Approximating Definite Integrals](https://tutorial.math.lamar.edu/Classes/CalcII/ApproximatingDefIntegrals.aspx) - [8\. Applications of Integrals](https://tutorial.math.lamar.edu/Classes/CalcII/IntAppsIntro.aspx) - [8\.1 Arc Length](https://tutorial.math.lamar.edu/Classes/CalcII/ArcLength.aspx) - [8\.2 Surface Area](https://tutorial.math.lamar.edu/Classes/CalcII/SurfaceArea.aspx) - [8\.3 Center of Mass](https://tutorial.math.lamar.edu/Classes/CalcII/CenterOfMass.aspx) - [8\.4 Hydrostatic Pressure](https://tutorial.math.lamar.edu/Classes/CalcII/HydrostaticPressure.aspx) - [8\.5 Probability](https://tutorial.math.lamar.edu/Classes/CalcII/Probability.aspx) - [9\. Parametric Equations and Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/ParametricIntro.aspx) - [9\.1 Parametric Equations and Curves](https://tutorial.math.lamar.edu/Classes/CalcII/ParametricEqn.aspx) - [9\.2 Tangents with Parametric Equations](https://tutorial.math.lamar.edu/Classes/CalcII/ParaTangent.aspx) - [9\.3 Area with Parametric Equations](https://tutorial.math.lamar.edu/Classes/CalcII/ParaArea.aspx) - [9\.4 Arc Length with Parametric Equations](https://tutorial.math.lamar.edu/Classes/CalcII/ParaArcLength.aspx) - [9\.5 Surface Area with Parametric Equations](https://tutorial.math.lamar.edu/Classes/CalcII/ParaSurfaceArea.aspx) - [9\.6 Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/PolarCoordinates.aspx) - [9\.7 Tangents with Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/PolarTangents.aspx) - [9\.8 Area with Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/PolarArea.aspx) - [9\.9 Arc Length with Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/PolarArcLength.aspx) - [9\.10 Surface Area with Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/PolarSurfaceArea.aspx) - [9\.11 Arc Length and Surface Area Revisited](https://tutorial.math.lamar.edu/Classes/CalcII/ArcLength_SurfaceArea.aspx) - [10\. Series & Sequences](https://tutorial.math.lamar.edu/Classes/CalcII/SeriesIntro.aspx) - [10\.1 Sequences](https://tutorial.math.lamar.edu/Classes/CalcII/Sequences.aspx) - [10\.2 More on Sequences](https://tutorial.math.lamar.edu/Classes/CalcII/MoreSequences.aspx) - [10\.3 Series - The Basics](https://tutorial.math.lamar.edu/Classes/CalcII/Series_Basics.aspx) - [10\.4 Convergence/Divergence of Series](https://tutorial.math.lamar.edu/Classes/CalcII/ConvergenceOfSeries.aspx) - [10\.5 Special Series](https://tutorial.math.lamar.edu/Classes/CalcII/Series_Special.aspx) - [10\.6 Integral Test](https://tutorial.math.lamar.edu/Classes/CalcII/IntegralTest.aspx) - [10\.7 Comparison Test/Limit Comparison Test](https://tutorial.math.lamar.edu/Classes/CalcII/SeriesCompTest.aspx) - [10\.8 Alternating Series Test](https://tutorial.math.lamar.edu/Classes/CalcII/AlternatingSeries.aspx) - [10\.9 Absolute Convergence](https://tutorial.math.lamar.edu/Classes/CalcII/AbsoluteConvergence.aspx) - [10\.10 Ratio Test](https://tutorial.math.lamar.edu/Classes/CalcII/RatioTest.aspx) - [10\.11 Root Test](https://tutorial.math.lamar.edu/Classes/CalcII/RootTest.aspx) - [10\.12 Strategy for Series](https://tutorial.math.lamar.edu/Classes/CalcII/SeriesStrategy.aspx) - [10\.13 Estimating the Value of a Series](https://tutorial.math.lamar.edu/Classes/CalcII/EstimatingSeries.aspx) - [10\.14 Power Series](https://tutorial.math.lamar.edu/Classes/CalcII/PowerSeries.aspx) - [10\.15 Power Series and Functions](https://tutorial.math.lamar.edu/Classes/CalcII/PowerSeriesandFunctions.aspx) - [10\.16 Taylor Series](https://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx) - [10\.17 Applications of Series](https://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeriesApps.aspx) - [10\.18 Binomial Series](https://tutorial.math.lamar.edu/Classes/CalcII/BinomialSeries.aspx) - [11\. Vectors](https://tutorial.math.lamar.edu/Classes/CalcII/VectorsIntro.aspx) - [11\.1 Vectors - The Basics](https://tutorial.math.lamar.edu/Classes/CalcII/Vectors_Basics.aspx) - [11\.2 Vector Arithmetic](https://tutorial.math.lamar.edu/Classes/CalcII/VectorArithmetic.aspx) - [11\.3 Dot Product](https://tutorial.math.lamar.edu/Classes/CalcII/DotProduct.aspx) - [11\.4 Cross Product](https://tutorial.math.lamar.edu/Classes/CalcII/CrossProduct.aspx) - [12\. 3-Dimensional Space](https://tutorial.math.lamar.edu/Classes/CalcII/3DSpace.aspx) - [12\.1 The 3-D Coordinate System](https://tutorial.math.lamar.edu/Classes/CalcII/3DCoords.aspx) - [12\.2 Equations of Lines](https://tutorial.math.lamar.edu/Classes/CalcII/EqnsOfLines.aspx) - [12\.3 Equations of Planes](https://tutorial.math.lamar.edu/Classes/CalcII/EqnsOfPlanes.aspx) - [12\.4 Quadric Surfaces](https://tutorial.math.lamar.edu/Classes/CalcII/QuadricSurfaces.aspx) - [12\.5 Functions of Several Variables](https://tutorial.math.lamar.edu/Classes/CalcII/MultiVrbleFcns.aspx) - [12\.6 Vector Functions](https://tutorial.math.lamar.edu/Classes/CalcII/VectorFunctions.aspx) - [12\.7 Calculus with Vector Functions](https://tutorial.math.lamar.edu/Classes/CalcII/VectorFcnsCalculus.aspx) - [12\.8 Tangent, Normal and Binormal Vectors](https://tutorial.math.lamar.edu/Classes/CalcII/TangentNormalVectors.aspx) - [12\.9 Arc Length with Vector Functions](https://tutorial.math.lamar.edu/Classes/CalcII/VectorArcLength.aspx) - [12\.10 Curvature](https://tutorial.math.lamar.edu/Classes/CalcII/Curvature.aspx) - [12\.11 Velocity and Acceleration](https://tutorial.math.lamar.edu/Classes/CalcII/Velocity_Acceleration.aspx) - [12\.12 Cylindrical Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/CylindricalCoords.aspx) - [12\.13 Spherical Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/SphericalCoords.aspx) - [Calculus III](https://tutorial.math.lamar.edu/Classes/CalcIII/CalcIII.aspx) - [12\. 3-Dimensional Space](https://tutorial.math.lamar.edu/Classes/CalcIII/3DSpace.aspx) - [12\.1 The 3-D Coordinate System](https://tutorial.math.lamar.edu/Classes/CalcIII/3DCoords.aspx) - [12\.2 Equations of Lines](https://tutorial.math.lamar.edu/Classes/CalcIII/EqnsOfLines.aspx) - [12\.3 Equations of Planes](https://tutorial.math.lamar.edu/Classes/CalcIII/EqnsOfPlanes.aspx) - [12\.4 Quadric Surfaces](https://tutorial.math.lamar.edu/Classes/CalcIII/QuadricSurfaces.aspx) - [12\.5 Functions of Several Variables](https://tutorial.math.lamar.edu/Classes/CalcIII/MultiVrbleFcns.aspx) - [12\.6 Vector Functions](https://tutorial.math.lamar.edu/Classes/CalcIII/VectorFunctions.aspx) - [12\.7 Calculus with Vector Functions](https://tutorial.math.lamar.edu/Classes/CalcIII/VectorFcnsCalculus.aspx) - [12\.8 Tangent, Normal and Binormal Vectors](https://tutorial.math.lamar.edu/Classes/CalcIII/TangentNormalVectors.aspx) - [12\.9 Arc Length with Vector Functions](https://tutorial.math.lamar.edu/Classes/CalcIII/VectorArcLength.aspx) - [12\.10 Curvature](https://tutorial.math.lamar.edu/Classes/CalcIII/Curvature.aspx) - [12\.11 Velocity and Acceleration](https://tutorial.math.lamar.edu/Classes/CalcIII/Velocity_Acceleration.aspx) - [12\.12 Cylindrical Coordinates](https://tutorial.math.lamar.edu/Classes/CalcIII/CylindricalCoords.aspx) - [12\.13 Spherical Coordinates](https://tutorial.math.lamar.edu/Classes/CalcIII/SphericalCoords.aspx) - [13\. Partial Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/PartialDerivsIntro.aspx) - [13\.1 Limits](https://tutorial.math.lamar.edu/Classes/CalcIII/Limits.aspx) - [13\.2 Partial Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/PartialDerivatives.aspx) - [13\.3 Interpretations of Partial Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/PartialDerivInterp.aspx) - [13\.4 Higher Order Partial Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/HighOrderPartialDerivs.aspx) - [13\.5 Differentials](https://tutorial.math.lamar.edu/Classes/CalcIII/Differentials.aspx) - [13\.6 Chain Rule](https://tutorial.math.lamar.edu/Classes/CalcIII/ChainRule.aspx) - [13\.7 Directional Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/DirectionalDeriv.aspx) - [14\. Applications of Partial Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/PartialDerivAppsIntro.aspx) - [14\.1 Tangent Planes and Linear Approximations](https://tutorial.math.lamar.edu/Classes/CalcIII/TangentPlanes.aspx) - [14\.2 Gradient Vector, Tangent Planes and Normal Lines](https://tutorial.math.lamar.edu/Classes/CalcIII/GradientVectorTangentPlane.aspx) - [14\.3 Relative Minimums and Maximums](https://tutorial.math.lamar.edu/Classes/CalcIII/RelativeExtrema.aspx) - [14\.4 Absolute Minimums and Maximums](https://tutorial.math.lamar.edu/Classes/CalcIII/AbsoluteExtrema.aspx) - [14\.5 Lagrange Multipliers](https://tutorial.math.lamar.edu/Classes/CalcIII/LagrangeMultipliers.aspx) - [15\. Multiple Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/MultipleIntegralsIntro.aspx) - [15\.1 Double Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/DoubleIntegrals.aspx) - [15\.2 Iterated Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/IteratedIntegrals.aspx) - [15\.3 Double Integrals over General Regions](https://tutorial.math.lamar.edu/Classes/CalcIII/DIGeneralRegion.aspx) - [15\.4 Double Integrals in Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcIII/DIPolarCoords.aspx) - [15\.5 Triple Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/TripleIntegrals.aspx) - [15\.6 Triple Integrals in Cylindrical Coordinates](https://tutorial.math.lamar.edu/Classes/CalcIII/TICylindricalCoords.aspx) - [15\.7 Triple Integrals in Spherical Coordinates](https://tutorial.math.lamar.edu/Classes/CalcIII/TISphericalCoords.aspx) - [15\.8 Change of Variables](https://tutorial.math.lamar.edu/Classes/CalcIII/ChangeOfVariables.aspx) - [15\.9 Surface Area](https://tutorial.math.lamar.edu/Classes/CalcIII/SurfaceArea.aspx) - [15\.10 Area and Volume Revisited](https://tutorial.math.lamar.edu/Classes/CalcIII/Area_Volume.aspx) - [16\. Line Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsIntro.aspx) - [16\.1 Vector Fields](https://tutorial.math.lamar.edu/Classes/CalcIII/VectorFields.aspx) - [16\.2 Line Integrals - Part I](https://tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsPtI.aspx) - [16\.3 Line Integrals - Part II](https://tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsPtII.aspx) - [16\.4 Line Integrals of Vector Fields](https://tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsVectorFields.aspx) - [16\.5 Fundamental Theorem for Line Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/FundThmLineIntegrals.aspx) - [16\.6 Conservative Vector Fields](https://tutorial.math.lamar.edu/Classes/CalcIII/ConservativeVectorField.aspx) - [16\.7 Green's Theorem](https://tutorial.math.lamar.edu/Classes/CalcIII/GreensTheorem.aspx) - [17\.Surface Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/SurfaceIntegralsIntro.aspx) - [17\.1 Curl and Divergence](https://tutorial.math.lamar.edu/Classes/CalcIII/CurlDivergence.aspx) - [17\.2 Parametric Surfaces](https://tutorial.math.lamar.edu/Classes/CalcIII/ParametricSurfaces.aspx) - [17\.3 Surface Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/SurfaceIntegrals.aspx) - [17\.4 Surface Integrals of Vector Fields](https://tutorial.math.lamar.edu/Classes/CalcIII/SurfIntVectorField.aspx) - [17\.5 Stokes' Theorem](https://tutorial.math.lamar.edu/Classes/CalcIII/StokesTheorem.aspx) - [17\.6 Divergence Theorem](https://tutorial.math.lamar.edu/Classes/CalcIII/DivergenceTheorem.aspx) - [Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/DE.aspx) - [1\. Basic Concepts](https://tutorial.math.lamar.edu/Classes/DE/IntroBasic.aspx) - [1\.1 Definitions](https://tutorial.math.lamar.edu/Classes/DE/Definitions.aspx) - [1\.2 Direction Fields](https://tutorial.math.lamar.edu/Classes/DE/DirectionFields.aspx) - [1\.3 Final Thoughts](https://tutorial.math.lamar.edu/Classes/DE/FinalThoughts.aspx) - [2\. First Order DE's](https://tutorial.math.lamar.edu/Classes/DE/IntroFirstOrder.aspx) - [2\.1 Linear Equations](https://tutorial.math.lamar.edu/Classes/DE/Linear.aspx) - [2\.2 Separable Equations](https://tutorial.math.lamar.edu/Classes/DE/Separable.aspx) - [2\.3 Exact Equations](https://tutorial.math.lamar.edu/Classes/DE/Exact.aspx) - [2\.4 Bernoulli Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/Bernoulli.aspx) - [2\.5 Substitutions](https://tutorial.math.lamar.edu/Classes/DE/Substitutions.aspx) - [2\.6 Intervals of Validity](https://tutorial.math.lamar.edu/Classes/DE/IoV.aspx) - [2\.7 Modeling with First Order DE's](https://tutorial.math.lamar.edu/Classes/DE/Modeling.aspx) - [2\.8 Equilibrium Solutions](https://tutorial.math.lamar.edu/Classes/DE/EquilibriumSolutions.aspx) - [2\.9 Euler's Method](https://tutorial.math.lamar.edu/Classes/DE/EulersMethod.aspx) - [3\. Second Order DE's](https://tutorial.math.lamar.edu/Classes/DE/IntroSecondOrder.aspx) - [3\.1 Basic Concepts](https://tutorial.math.lamar.edu/Classes/DE/SecondOrderConcepts.aspx) - [3\.2 Real & Distinct Roots](https://tutorial.math.lamar.edu/Classes/DE/RealRoots.aspx) - [3\.3 Complex Roots](https://tutorial.math.lamar.edu/Classes/DE/ComplexRoots.aspx) - [3\.4 Repeated Roots](https://tutorial.math.lamar.edu/Classes/DE/RepeatedRoots.aspx) - [3\.5 Reduction of Order](https://tutorial.math.lamar.edu/Classes/DE/ReductionofOrder.aspx) - [3\.6 Fundamental Sets of Solutions](https://tutorial.math.lamar.edu/Classes/DE/FundamentalSetsofSolutions.aspx) - [3\.7 More on the Wronskian](https://tutorial.math.lamar.edu/Classes/DE/Wronskian.aspx) - [3\.8 Nonhomogeneous Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/NonhomogeneousDE.aspx) - [3\.9 Undetermined Coefficients](https://tutorial.math.lamar.edu/Classes/DE/UndeterminedCoefficients.aspx) - [3\.10 Variation of Parameters](https://tutorial.math.lamar.edu/Classes/DE/VariationofParameters.aspx) - [3\.11 Mechanical Vibrations](https://tutorial.math.lamar.edu/Classes/DE/Vibrations.aspx) - [4\. Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/LaplaceIntro.aspx) - [4\.1 The Definition](https://tutorial.math.lamar.edu/Classes/DE/LaplaceDefinition.aspx) - [4\.2 Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/LaplaceTransforms.aspx) - [4\.3 Inverse Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/InverseTransforms.aspx) - [4\.4 Step Functions](https://tutorial.math.lamar.edu/Classes/DE/StepFunctions.aspx) - [4\.5 Solving IVP's with Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/IVPWithLaplace.aspx) - [4\.6 Nonconstant Coefficient IVP's](https://tutorial.math.lamar.edu/Classes/DE/IVPWithNonConstantCoefficient.aspx) - [4\.7 IVP's With Step Functions](https://tutorial.math.lamar.edu/Classes/DE/IVPWithStepFunction.aspx) - [4\.8 Dirac Delta Function](https://tutorial.math.lamar.edu/Classes/DE/DiracDeltaFunction.aspx) - [4\.9 Convolution Integrals](https://tutorial.math.lamar.edu/Classes/DE/ConvolutionIntegrals.aspx) - [4\.10 Table Of Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/Laplace_Table.aspx) - [5\. Systems of DE's](https://tutorial.math.lamar.edu/Classes/DE/SystemsIntro.aspx) - [5\.1 Review : Systems of Equations](https://tutorial.math.lamar.edu/Classes/DE/LA_Systems.aspx) - [5\.2 Review : Matrices & Vectors](https://tutorial.math.lamar.edu/Classes/DE/LA_Matrix.aspx) - [5\.3 Review : Eigenvalues & Eigenvectors](https://tutorial.math.lamar.edu/Classes/DE/LA_Eigen.aspx) - [5\.4 Systems of Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/SystemsDE.aspx) - [5\.5 Solutions to Systems](https://tutorial.math.lamar.edu/Classes/DE/SolutionsToSystems.aspx) - [5\.6 Phase Plane](https://tutorial.math.lamar.edu/Classes/DE/PhasePlane.aspx) - [5\.7 Real Eigenvalues](https://tutorial.math.lamar.edu/Classes/DE/RealEigenvalues.aspx) - [5\.8 Complex Eigenvalues](https://tutorial.math.lamar.edu/Classes/DE/ComplexEigenvalues.aspx) - [5\.9 Repeated Eigenvalues](https://tutorial.math.lamar.edu/Classes/DE/RepeatedEigenvalues.aspx) - [5\.10 Nonhomogeneous Systems](https://tutorial.math.lamar.edu/Classes/DE/NonhomogeneousSystems.aspx) - [5\.11 Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/SystemsLaplace.aspx) - [5\.12 Modeling](https://tutorial.math.lamar.edu/Classes/DE/SystemsModeling.aspx) - [6\. Series Solutions to DE's](https://tutorial.math.lamar.edu/Classes/DE/SeriesIntro.aspx) - [6\.1 Review : Power Series](https://tutorial.math.lamar.edu/Classes/DE/PowerSeries.aspx) - [6\.2 Review : Taylor Series](https://tutorial.math.lamar.edu/Classes/DE/TaylorSeries.aspx) - [6\.3 Series Solutions](https://tutorial.math.lamar.edu/Classes/DE/SeriesSolutions.aspx) - [6\.4 Euler Equations](https://tutorial.math.lamar.edu/Classes/DE/EulerEquations.aspx) - [7\. Higher Order Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/IntroHigherOrder.aspx) - [7\.1 Basic Concepts for nth Order Linear Equations](https://tutorial.math.lamar.edu/Classes/DE/HOBasicConcepts.aspx) - [7\.2 Linear Homogeneous Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/HOHomogeneousDE.aspx) - [7\.3 Undetermined Coefficients](https://tutorial.math.lamar.edu/Classes/DE/HOUndeterminedCoeff.aspx) - [7\.4 Variation of Parameters](https://tutorial.math.lamar.edu/Classes/DE/HOVariationOfParam.aspx) - [7\.5 Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/HOLaplaceTransforms.aspx) - [7\.6 Systems of Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/HOSystems.aspx) - [7\.7 Series Solutions](https://tutorial.math.lamar.edu/Classes/DE/HOSeries.aspx) - [8\. Boundary Value Problems & Fourier Series](https://tutorial.math.lamar.edu/Classes/DE/IntroBVP.aspx) - [8\.1 Boundary Value Problems](https://tutorial.math.lamar.edu/Classes/DE/BoundaryValueProblem.aspx) - [8\.2 Eigenvalues and Eigenfunctions](https://tutorial.math.lamar.edu/Classes/DE/BVPEvals.aspx) - [8\.3 Periodic Functions & Orthogonal Functions](https://tutorial.math.lamar.edu/Classes/DE/PeriodicOrthogonal.aspx) - [8\.4 Fourier Sine Series](https://tutorial.math.lamar.edu/Classes/DE/FourierSineSeries.aspx) - [8\.5 Fourier Cosine Series](https://tutorial.math.lamar.edu/Classes/DE/FourierCosineSeries.aspx) - [8\.6 Fourier Series](https://tutorial.math.lamar.edu/Classes/DE/FourierSeries.aspx) - [8\.7 Convergence of Fourier Series](https://tutorial.math.lamar.edu/Classes/DE/ConvergenceFourierSeries.aspx) - [9\. Partial Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/IntroPDE.aspx) - [9\.1 The Heat Equation](https://tutorial.math.lamar.edu/Classes/DE/TheHeatEquation.aspx) - [9\.2 The Wave Equation](https://tutorial.math.lamar.edu/Classes/DE/TheWaveEquation.aspx) - [9\.3 Terminology](https://tutorial.math.lamar.edu/Classes/DE/PDETerminology.aspx) - [9\.4 Separation of Variables](https://tutorial.math.lamar.edu/Classes/DE/SeparationofVariables.aspx) - [9\.5 Solving the Heat Equation](https://tutorial.math.lamar.edu/Classes/DE/SolvingHeatEquation.aspx) - [9\.6 Heat Equation with Non-Zero Temperature Boundaries](https://tutorial.math.lamar.edu/Classes/DE/HeatEqnNonZero.aspx) - [9\.7 Laplace's Equation](https://tutorial.math.lamar.edu/Classes/DE/LaplacesEqn.aspx) - [9\.8 Vibrating String](https://tutorial.math.lamar.edu/Classes/DE/VibratingString.aspx) - [9\.9 Summary of Separation of Variables](https://tutorial.math.lamar.edu/Classes/DE/PDESummary.aspx) - Extras - [Algebra & Trig Review](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/AlgebraTrig.aspx) - [1\. Algebra](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/AlgebraIntro.aspx) - [1\.1 Exponents](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/Exponents.aspx) - [1\.2 Absolute Value](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/AbsoluteValue.aspx) - [1\.3 Radicals](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/Radicals.aspx) - [1\.4 Rationalizing](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/Rationalizing.aspx) - [1\.5 Functions](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/Functions.aspx) - [1\.6 Multiplying Polynomials](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/MultPoly.aspx) - [1\.7 Factoring](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/Factoring.aspx) - [1\.8 Simplifying Rational Expressions](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SimpRatExp.aspx) - [1\.9 Graphing and Common Graphs](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/Graphing.aspx) - [1\.10 Solving Equations, Part I](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SolveEqnPtI.aspx) - [1\.11 Solving Equations, Part II](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SolveEqnPtII.aspx) - [1\.12 Solving Systems of Equations](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SolveSystems.aspx) - [1\.13 Solving Inequalities](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SolveIneq.aspx) - [1\.14 Absolute Value Equations and Inequalities](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SolveAbsValue.aspx) - [2\. Trigonometry](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/TrigIntro.aspx) - [2\.1 Trig Function Evaluation](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/TrigFunctions.aspx) - [2\.2 Graphs of Trig Functions](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/TrigGraphs.aspx) - [2\.3 Trig Formulas](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/TrigFormulas.aspx) - [2\.4 Solving Trig Equations](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SolveTrigEqn.aspx) - [2\.5 Inverse Trig Functions](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/InverseTrig.aspx) - [3\. Exponentials & Logarithms](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/ExpLogIntro.aspx) - [3\.1 Basic Exponential Functions](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/ExponentialFcns.aspx) - [3\.2 Basic Logarithm Functions](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/LogarithmFcns.aspx) - [3\.3 Logarithm Properties](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/LogProperties.aspx) - [3\.4 Simplifying Logarithms](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SimpLogs.aspx) - [3\.5 Solving Exponential Equations](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SolveExpEqn.aspx) - [3\.6 Solving Logarithm Equations](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SolveLogEqn.aspx) - [Common Math Errors](https://tutorial.math.lamar.edu/Extras/CommonErrors/CommonMathErrors.aspx) - [1\. General Errors](https://tutorial.math.lamar.edu/Extras/CommonErrors/GeneralErrors.aspx) - [2\. Algebra Errors](https://tutorial.math.lamar.edu/Extras/CommonErrors/AlgebraErrors.aspx) - [3\. Trig Errors](https://tutorial.math.lamar.edu/Extras/CommonErrors/TrigErrors.aspx) - [4\. Common Errors](https://tutorial.math.lamar.edu/Extras/CommonErrors/CommonErrors.aspx) - [5\. Calculus Errors](https://tutorial.math.lamar.edu/Extras/CommonErrors/CalculusErrors.aspx) - [Complex Number Primer](https://tutorial.math.lamar.edu/Extras/ComplexPrimer/ComplexNumbers.aspx) - [1\. The Definition](https://tutorial.math.lamar.edu/Extras/ComplexPrimer/Definition.aspx) - [2\. Arithmetic](https://tutorial.math.lamar.edu/Extras/ComplexPrimer/Arithmetic.aspx) - [3\. Conjugate and Modulus](https://tutorial.math.lamar.edu/Extras/ComplexPrimer/ConjugateModulus.aspx) - [4\. Polar and Exponential Forms](https://tutorial.math.lamar.edu/Extras/ComplexPrimer/Forms.aspx) - [5\. 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Learn From Your Errors](https://tutorial.math.lamar.edu/Extras/StudyMath/Errors.aspx) - [Cheat Sheets & Tables](https://tutorial.math.lamar.edu/Extras/CheatSheets_Tables.aspx) - Misc Links - [Contact Me](https://tutorial.math.lamar.edu/contact.aspx) - [Links](https://tutorial.math.lamar.edu/links.aspx) - [MathJax Help and Configuration](https://tutorial.math.lamar.edu/mathjax.aspx) - [Privacy Statement](https://tutorial.math.lamar.edu/privacy.aspx) - [Site Help & FAQ](https://tutorial.math.lamar.edu/help.aspx) - [Terms of Use](https://tutorial.math.lamar.edu/terms.aspx) Paul's Online Notes [Home](https://tutorial.math.lamar.edu/ "Go to Main Page") / [Calculus III](https://tutorial.math.lamar.edu/Classes/CalcIII/CalcIII.aspx "Go to Book Introduction") / [Partial Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/PartialDerivsIntro.aspx "Go to Chapter Introduction") / Higher Order Partial Derivatives [Prev. Section](https://tutorial.math.lamar.edu/Classes/CalcIII/PartialDerivInterp.aspx "Goto Previous Section : Interpretations of Partial Derivatives") Notes [Practice Problems](https://tutorial.math.lamar.edu/Problems/CalcIII/HighOrderPartialDerivs.aspx "Go to Practice Problems for current topic.") [Assignment Problems](https://tutorial.math.lamar.edu/ProblemsNS/CalcIII/HighOrderPartialDerivs.aspx "Go to Assignment Problems for current topic.") [Next Section](https://tutorial.math.lamar.edu/Classes/CalcIII/Differentials.aspx "Goto Next Section : Differentials") Show Mobile Notice Show All Notes Hide All Notes Mobile Notice You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width. ### Section 13.4 : Higher Order Partial Derivatives Just as we had higher order derivatives with functions of one variable we will also have higher order derivatives of functions of more than one variable. However, this time we will have more options since we do have more than one variable. Consider the case of a function of two variables, \\(f\\left( {x,y} \\right)\\) since both of the first order partial derivatives are also functions of \\(x\\) and \\(y\\) we could in turn differentiate each with respect to \\(x\\) or \\(y\\). This means that for the case of a function of two variables there will be a total of four possible second order derivatives. Here they are and the notations that we’ll use to denote them. \\\[\\begin{align\}{\\left( {{f\_x}} \\right)\_x} & = {f\_{x\\,x}} = \\frac{\\partial }{{\\partial x}}\\left( {\\frac{{\\partial f}}{{\\partial x}}} \\right) = \\frac{{{\\partial ^2}f}}{{\\partial {x^2}}}\\\\ {\\left( {{f\_x}} \\right)\_y} & = {f\_{x\\,y}} = \\frac{\\partial }{{\\partial y}}\\left( {\\frac{{\\partial f}}{{\\partial x}}} \\right) = \\frac{{{\\partial ^2}f}}{{\\partial y\\partial x}}\\\\ {\\left( {{f\_y}} \\right)\_x} & = {f\_{y\\,x}} = \\frac{\\partial }{{\\partial x}}\\left( {\\frac{{\\partial f}}{{\\partial y}}} \\right) = \\frac{{{\\partial ^2}f}}{{\\partial x\\partial y}}\\\\ {\\left( {{f\_y}} \\right)\_y} & = {f\_{y\\,y}} = \\frac{\\partial }{{\\partial y}}\\left( {\\frac{{\\partial f}}{{\\partial y}}} \\right) = \\frac{{{\\partial ^2}f}}{{\\partial {y^2}}}\\end{align\}\\\] The second and third second order partial derivatives are often called mixed partial derivatives since we are taking derivatives with respect to more than one variable. Note as well that the order that we take the derivatives in is given by the notation for each these. If we are using the subscripting notation, e.g. \\({f\_{x\\,y}}\\), then we will differentiate from left to right. In other words, in this case, we will differentiate first with respect to \\(x\\) and then with respect to \\(y\\). With the fractional notation, e.g. \\(\\frac{{{\\partial ^2}f}}{{\\partial y\\partial x}}\\), it is the opposite. In these cases we differentiate moving along the denominator from right to left. So, again, in this case we differentiate with respect to \\(x\\) first and then \\(y\\). Let’s take a quick look at an example. Example 1 Find all the second order derivatives for \\(f\\left( {x,y} \\right) = \\cos \\left( {2x} \\right) - {x^2}{{\\bf{e}}^{5y}} + 3{y^2}\\). Show Solution We’ll first need the first order derivatives so here they are. \\\[\\begin{align\}{f\_x}\\left( {x,y} \\right) & = - 2\\sin \\left( {2x} \\right) - 2x{{\\bf{e}}^{5y}}\\\\ {f\_y}\\left( {x,y} \\right) & = - 5{x^2}{{\\bf{e}}^{5y}} + 6y\\end{align\}\\\] Now, let’s get the second order derivatives. \\\[\\begin{align\}{f\_{xx}} & = - 4\\cos \\left( {2x} \\right) - 2{{\\bf{e}}^{5y}}\\\\ {f\_{xy}} & = - 10x{{\\bf{e}}^{5y}}\\\\ {f\_{yx}} & = - 10x{{\\bf{e}}^{5y}}\\\\ {f\_{yy}} & = - 25{x^2}{{\\bf{e}}^{5y}} + 6\\end{align\}\\\] Notice that we dropped the \\(\\left( {x,y} \\right)\\) from the derivatives. This is fairly standard and we will be doing it most of the time from this point on. We will also be dropping it for the first order derivatives in most cases. Now let’s also notice that, in this case, \\({f\_{xy}} = {f\_{yx}}\\). This is not by coincidence. If the function is “nice enough” this will always be the case. So, what’s “nice enough”? The following theorem tells us. #### Clairaut’s Theorem Suppose that \\(f\\) is defined on a disk \\(D\\) that contains the point \\(\\left( {a,b} \\right)\\). If the functions \\({f\_{xy}}\\) and \\({f\_{yx}}\\) are continuous on this disk then, \\\[{f\_{xy}}\\left( {a,b} \\right) = {f\_{yx}}\\left( {a,b} \\right)\\\] Now, do not get too excited about the disk business and the fact that we gave the theorem for a specific point. In pretty much every example in this class if the two mixed second order partial derivatives are continuous then they will be equal. Example 2 Verify Clairaut’s Theorem for \\(f\\left( {x,y} \\right) = x{{\\bf{e}}^{ - {x^2}{y^2}}}\\). Show Solution We’ll first need the two first order derivatives. \\\[\\begin{align\}{f\_x}\\left( {x,y} \\right) & = {{\\bf{e}}^{ - {x^2}{y^2}}} - 2{x^2}{y^2}{{\\bf{e}}^{ - {x^2}{y^2}}}\\\\ {f\_y}\\left( {x,y} \\right) & = - 2y{x^3}{{\\bf{e}}^{ - {x^2}{y^2}}}\\end{align\}\\\] Now, compute the two mixed second order partial derivatives. \\\[\\begin{align\}{f\_{xy}}\\left( {x,y} \\right) & = - 2y{x^2}{{\\bf{e}}^{ - {x^2}{y^2}}} - 4{x^2}y{{\\bf{e}}^{ - {x^2}{y^2}}} + 4{x^4}{y^3}{{\\bf{e}}^{ - {x^2}{y^2}}} = - 6{x^2}y{{\\bf{e}}^{ - {x^2}{y^2}}} + 4{x^4}{y^3}{{\\bf{e}}^{ - {x^2}{y^2}}}\\\\ {f\_{yx}}\\left( {x,y} \\right) & = - 6y{x^2}{{\\bf{e}}^{ - {x^2}{y^2}}} + 4{y^3}{x^4}{{\\bf{e}}^{ - {x^2}{y^2}}}\\end{align\}\\\] Sure enough they are the same. So far we have only looked at second order derivatives. There are, of course, higher order derivatives as well. Here are a couple of the third order partial derivatives of function of two variables. \\\[\\begin{align\}{f\_{x\\,y\\,x}} & = {\\left( {{f\_{xy}}} \\right)\_x} = \\frac{\\partial }{{\\partial x}}\\left( {\\frac{{{\\partial ^2}f}}{{\\partial y\\partial x}}} \\right) = \\frac{{{\\partial ^3}f}}{{\\partial x\\partial y\\partial x}}\\\\ {f\_{y\\,x\\,x}} & = {\\left( {{f\_{yx}}} \\right)\_x} = \\frac{\\partial }{{\\partial x}}\\left( {\\frac{{{\\partial ^2}f}}{{\\partial x\\partial y}}} \\right) = \\frac{{{\\partial ^3}f}}{{\\partial {x^2}\\partial y}}\\end{align\}\\\] Notice as well that for both of these we differentiate once with respect to \\(y\\) and twice with respect to \\(x\\). There is also another third order partial derivative in which we can do this, \\({f\_{x\\,x\\,y}}\\). There is an extension to Clairaut’s Theorem that says if all three of these are continuous then they should all be equal, \\\[{f\_{x\\,x\\,y}} = {f\_{x\\,y\\,x}} = {f\_{y\\,x\\,x}}\\\] To this point we’ve only looked at functions of two variables, but everything that we’ve done to this point will work regardless of the number of variables that we’ve got in the function and there are natural extensions to Clairaut’s theorem to all of these cases as well. For instance, \\\[{f\_{x\\,z}}\\left( {x,y,z} \\right) = {f\_{z\\,x}}\\left( {x,y,z} \\right)\\\] provided both of the derivatives are continuous. In general, we can extend Clairaut’s theorem to any function and mixed partial derivatives. The only requirement is that in each derivative we differentiate with respect to each variable the same number of times. In other words, provided we meet the continuity condition, the following will be equal \\\[{f\_{s\\,s\\,r\\,t\\,s\\,r\\,r}} = {f\_{t\\,r\\,s\\,r\\,s\\,s\\,r}}\\\] because in each case we differentiate with respect to \\(t\\) once, \\(s\\) three times and \\(r\\) three times. Let’s do a couple of examples with higher (well higher order than two anyway) order derivatives and functions of more than two variables. Example 3 Find the indicated derivative for each of the following functions. 1. Find \\({f\_{x\\,x\\,y\\,z\\,z}}\\) for \\(f\\left( {x,y,z} \\right) = {z^3}{y^2}\\ln \\left( x \\right)\\) 2. Find \\(\\displaystyle \\frac{{{\\partial ^3}f}}{{\\partial y\\partial {x^2}}}\\) for \\(f\\left( {x,y} \\right) = {{\\bf{e}}^{xy}}\\) Show All Solutions Hide All Solutions a Find \\({f\_{x\\,x\\,y\\,z\\,z}}\\) for \\(f\\left( {x,y,z} \\right) = {z^3}{y^2}\\ln \\left( x \\right)\\) Show Solution In this case remember that we differentiate from left to right. Here are the derivatives for this part. \\\[{f\_x} = \\frac{{{z^3}{y^2}}}{x}\\\] \\\[{f\_{xx}} = - \\frac{{{z^3}{y^2}}}{{{x^2}}}\\\] \\\[{f\_{xxy}} = - \\frac{{2{z^3}y}}{{{x^2}}}\\\] \\\[{f\_{xxyz}} = - \\frac{{6{z^2}y}}{{{x^2}}}\\\] \\\[{f\_{xxyzz}} = - \\frac{{12zy}}{{{x^2}}}\\\] b Find \\(\\displaystyle \\frac{{{\\partial ^3}f}}{{\\partial y\\partial {x^2}}}\\) for \\(f\\left( {x,y} \\right) = {{\\bf{e}}^{xy}}\\) Show Solution Here we differentiate from right to left. Here are the derivatives for this function. \\\[\\frac{{\\partial f}}{{\\partial x}} = y{{\\bf{e}}^{xy}}\\\] \\\[\\frac{{{\\partial ^2}f}}{{\\partial {x^2}}} = {y^2}{{\\bf{e}}^{xy}}\\\] \\\[\\frac{{{\\partial ^3}f}}{{\\partial y\\partial {x^2}}} = 2y{{\\bf{e}}^{xy}} + x{y^2}{{\\bf{e}}^{xy}}\\\] \[[Contact Me](https://tutorial.math.lamar.edu/Contact.aspx)\] \[[Privacy Statement](https://tutorial.math.lamar.edu/Privacy.aspx)\] \[[Site Help & FAQ](https://tutorial.math.lamar.edu/Help.aspx)\] \[[Terms of Use](https://tutorial.math.lamar.edu/Terms.aspx)\] © 2003 - 2026 Paul Dawkins Page Last Modified : 11/16/2022
Readable Markdown	Show Mobile Notice Show All Notes Hide All Notes Mobile Notice You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width. ### Section 13.4 : Higher Order Partial Derivatives Just as we had higher order derivatives with functions of one variable we will also have higher order derivatives of functions of more than one variable. However, this time we will have more options since we do have more than one variable. Consider the case of a function of two variables, \\(f\\left( {x,y} \\right)\\) since both of the first order partial derivatives are also functions of \\(x\\) and \\(y\\) we could in turn differentiate each with respect to \\(x\\) or \\(y\\). This means that for the case of a function of two variables there will be a total of four possible second order derivatives. Here they are and the notations that we’ll use to denote them. \\\[\\begin{align\}{\\left( {{f\_x}} \\right)\_x} & = {f\_{x\\,x}} = \\frac{\\partial }{{\\partial x}}\\left( {\\frac{{\\partial f}}{{\\partial x}}} \\right) = \\frac{{{\\partial ^2}f}}{{\\partial {x^2}}}\\\\ {\\left( {{f\_x}} \\right)\_y} & = {f\_{x\\,y}} = \\frac{\\partial }{{\\partial y}}\\left( {\\frac{{\\partial f}}{{\\partial x}}} \\right) = \\frac{{{\\partial ^2}f}}{{\\partial y\\partial x}}\\\\ {\\left( {{f\_y}} \\right)\_x} & = {f\_{y\\,x}} = \\frac{\\partial }{{\\partial x}}\\left( {\\frac{{\\partial f}}{{\\partial y}}} \\right) = \\frac{{{\\partial ^2}f}}{{\\partial x\\partial y}}\\\\ {\\left( {{f\_y}} \\right)\_y} & = {f\_{y\\,y}} = \\frac{\\partial }{{\\partial y}}\\left( {\\frac{{\\partial f}}{{\\partial y}}} \\right) = \\frac{{{\\partial ^2}f}}{{\\partial {y^2}}}\\end{align\}\\\] The second and third second order partial derivatives are often called mixed partial derivatives since we are taking derivatives with respect to more than one variable. Note as well that the order that we take the derivatives in is given by the notation for each these. If we are using the subscripting notation, e.g. \\({f\_{x\\,y}}\\), then we will differentiate from left to right. In other words, in this case, we will differentiate first with respect to \\(x\\) and then with respect to \\(y\\). With the fractional notation, e.g. \\(\\frac{{{\\partial ^2}f}}{{\\partial y\\partial x}}\\), it is the opposite. In these cases we differentiate moving along the denominator from right to left. So, again, in this case we differentiate with respect to \\(x\\) first and then \\(y\\). Let’s take a quick look at an example. Example 1 Find all the second order derivatives for \\(f\\left( {x,y} \\right) = \\cos \\left( {2x} \\right) - {x^2}{{\\bf{e}}^{5y}} + 3{y^2}\\). Show Solution We’ll first need the first order derivatives so here they are. \\\[\\begin{align\}{f\_x}\\left( {x,y} \\right) & = - 2\\sin \\left( {2x} \\right) - 2x{{\\bf{e}}^{5y}}\\\\ {f\_y}\\left( {x,y} \\right) & = - 5{x^2}{{\\bf{e}}^{5y}} + 6y\\end{align\}\\\] Now, let’s get the second order derivatives. \\\[\\begin{align\}{f\_{xx}} & = - 4\\cos \\left( {2x} \\right) - 2{{\\bf{e}}^{5y}}\\\\ {f\_{xy}} & = - 10x{{\\bf{e}}^{5y}}\\\\ {f\_{yx}} & = - 10x{{\\bf{e}}^{5y}}\\\\ {f\_{yy}} & = - 25{x^2}{{\\bf{e}}^{5y}} + 6\\end{align\}\\\] Notice that we dropped the \\(\\left( {x,y} \\right)\\) from the derivatives. This is fairly standard and we will be doing it most of the time from this point on. We will also be dropping it for the first order derivatives in most cases. Now let’s also notice that, in this case, \\({f\_{xy}} = {f\_{yx}}\\). This is not by coincidence. If the function is “nice enough” this will always be the case. So, what’s “nice enough”? The following theorem tells us. #### Clairaut’s Theorem Suppose that \\(f\\) is defined on a disk \\(D\\) that contains the point \\(\\left( {a,b} \\right)\\). If the functions \\({f\_{xy}}\\) and \\({f\_{yx}}\\) are continuous on this disk then, \\\[{f\_{xy}}\\left( {a,b} \\right) = {f\_{yx}}\\left( {a,b} \\right)\\\] Now, do not get too excited about the disk business and the fact that we gave the theorem for a specific point. In pretty much every example in this class if the two mixed second order partial derivatives are continuous then they will be equal. Example 2 Verify Clairaut’s Theorem for \\(f\\left( {x,y} \\right) = x{{\\bf{e}}^{ - {x^2}{y^2}}}\\). Show Solution We’ll first need the two first order derivatives. \\\[\\begin{align\}{f\_x}\\left( {x,y} \\right) & = {{\\bf{e}}^{ - {x^2}{y^2}}} - 2{x^2}{y^2}{{\\bf{e}}^{ - {x^2}{y^2}}}\\\\ {f\_y}\\left( {x,y} \\right) & = - 2y{x^3}{{\\bf{e}}^{ - {x^2}{y^2}}}\\end{align\}\\\] Now, compute the two mixed second order partial derivatives. \\\[\\begin{align\}{f\_{xy}}\\left( {x,y} \\right) & = - 2y{x^2}{{\\bf{e}}^{ - {x^2}{y^2}}} - 4{x^2}y{{\\bf{e}}^{ - {x^2}{y^2}}} + 4{x^4}{y^3}{{\\bf{e}}^{ - {x^2}{y^2}}} = - 6{x^2}y{{\\bf{e}}^{ - {x^2}{y^2}}} + 4{x^4}{y^3}{{\\bf{e}}^{ - {x^2}{y^2}}}\\\\ {f\_{yx}}\\left( {x,y} \\right) & = - 6y{x^2}{{\\bf{e}}^{ - {x^2}{y^2}}} + 4{y^3}{x^4}{{\\bf{e}}^{ - {x^2}{y^2}}}\\end{align\}\\\] Sure enough they are the same. So far we have only looked at second order derivatives. There are, of course, higher order derivatives as well. Here are a couple of the third order partial derivatives of function of two variables. \\\[\\begin{align\}{f\_{x\\,y\\,x}} & = {\\left( {{f\_{xy}}} \\right)\_x} = \\frac{\\partial }{{\\partial x}}\\left( {\\frac{{{\\partial ^2}f}}{{\\partial y\\partial x}}} \\right) = \\frac{{{\\partial ^3}f}}{{\\partial x\\partial y\\partial x}}\\\\ {f\_{y\\,x\\,x}} & = {\\left( {{f\_{yx}}} \\right)\_x} = \\frac{\\partial }{{\\partial x}}\\left( {\\frac{{{\\partial ^2}f}}{{\\partial x\\partial y}}} \\right) = \\frac{{{\\partial ^3}f}}{{\\partial {x^2}\\partial y}}\\end{align\}\\\] Notice as well that for both of these we differentiate once with respect to \\(y\\) and twice with respect to \\(x\\). There is also another third order partial derivative in which we can do this, \\({f\_{x\\,x\\,y}}\\). There is an extension to Clairaut’s Theorem that says if all three of these are continuous then they should all be equal, \\\[{f\_{x\\,x\\,y}} = {f\_{x\\,y\\,x}} = {f\_{y\\,x\\,x}}\\\] To this point we’ve only looked at functions of two variables, but everything that we’ve done to this point will work regardless of the number of variables that we’ve got in the function and there are natural extensions to Clairaut’s theorem to all of these cases as well. For instance, \\\[{f\_{x\\,z}}\\left( {x,y,z} \\right) = {f\_{z\\,x}}\\left( {x,y,z} \\right)\\\] provided both of the derivatives are continuous. In general, we can extend Clairaut’s theorem to any function and mixed partial derivatives. The only requirement is that in each derivative we differentiate with respect to each variable the same number of times. In other words, provided we meet the continuity condition, the following will be equal \\\[{f\_{s\\,s\\,r\\,t\\,s\\,r\\,r}} = {f\_{t\\,r\\,s\\,r\\,s\\,s\\,r}}\\\] because in each case we differentiate with respect to \\(t\\) once, \\(s\\) three times and \\(r\\) three times. Let’s do a couple of examples with higher (well higher order than two anyway) order derivatives and functions of more than two variables. Example 3 Find the indicated derivative for each of the following functions. 1. Find \\({f\_{x\\,x\\,y\\,z\\,z}}\\) for \\(f\\left( {x,y,z} \\right) = {z^3}{y^2}\\ln \\left( x \\right)\\) 2. Find \\(\\displaystyle \\frac{{{\\partial ^3}f}}{{\\partial y\\partial {x^2}}}\\) for \\(f\\left( {x,y} \\right) = {{\\bf{e}}^{xy}}\\) Show All Solutions Hide All Solutions a Find \\({f\_{x\\,x\\,y\\,z\\,z}}\\) for \\(f\\left( {x,y,z} \\right) = {z^3}{y^2}\\ln \\left( x \\right)\\) Show Solution In this case remember that we differentiate from left to right. Here are the derivatives for this part. \\\[{f\_x} = \\frac{{{z^3}{y^2}}}{x}\\\] \\\[{f\_{xx}} = - \\frac{{{z^3}{y^2}}}{{{x^2}}}\\\] \\\[{f\_{xxy}} = - \\frac{{2{z^3}y}}{{{x^2}}}\\\] \\\[{f\_{xxyz}} = - \\frac{{6{z^2}y}}{{{x^2}}}\\\] \\\[{f\_{xxyzz}} = - \\frac{{12zy}}{{{x^2}}}\\\] b Find \\(\\displaystyle \\frac{{{\\partial ^3}f}}{{\\partial y\\partial {x^2}}}\\) for \\(f\\left( {x,y} \\right) = {{\\bf{e}}^{xy}}\\) Show Solution Here we differentiate from right to left. Here are the derivatives for this function. \\\[\\frac{{\\partial f}}{{\\partial x}} = y{{\\bf{e}}^{xy}}\\\] \\\[\\frac{{{\\partial ^2}f}}{{\\partial {x^2}}} = {y^2}{{\\bf{e}}^{xy}}\\\] \\\[\\frac{{{\\partial ^3}f}}{{\\partial y\\partial {x^2}}} = 2y{{\\bf{e}}^{xy}} + x{y^2}{{\\bf{e}}^{xy}}\\\]
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