Calculus I - Product and Quotient Rule

<!DOCTYPE html> <html> <head><meta charset="utf-8" /><meta name="viewport" content="width=device-width, initial-scale=1, user-scalable=yes" /><meta http-equiv="X-UA-Compatible" content="IE=edge" />  <meta name="google-site-verification" content="uLoA31CJfOhIVMJWBjCmQL8xNMmmLybZU3LRKavy9WQ" /><title> Calculus I - Product and Quotient Rule </title>  <script async src="https://www.googletagmanager.com/gtag/js?id=G-9SCXJM7BEJ"></script> <script> window.dataLayer = window.dataLayer || []; function gtag() { dataLayer.push(arguments); } gtag('js', new Date()); gtag('config', 'G-9SCXJM7BEJ'); </script> <link type="text/css" href="/css/jquery.mmenu.all.css" rel="stylesheet" /><link type="text/css" href="/css/jquery.dropdown.css" rel="stylesheet" /><link href="/FA/css/all.min.css" rel="stylesheet" /><link type="text/css" href="/css/notes-all.css" rel="stylesheet" /><link type="text/css" href="/css/notes-google.css" rel="stylesheet" /><link type="text/css" href="/css/notes-mmenu.css" rel="stylesheet" /><link type="text/css" href="/css/notes-dropdown.css" rel="stylesheet" /> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ TeX: { equationNumbers: { autoNumber: "AMS" } } }); </script> <script type="text/javascript" async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/MathJax.js?config=TeX-AMS_CHTML-full"></script> <script type="text/javascript" src="/js/jquery_on.js"></script> <script type="text/javascript" src="/js/jquery.mmenu.all.js"></script> <script type="text/javascript" src="/js/jquery.dropdown.js"></script> <script type="text/javascript" src="/js/notes-all.js"></script> <script> (function () { var cx = '001004262401526223570:11yv6vpcqvy'; var gcse = document.createElement('script'); gcse.type = 'text/javascript'; gcse.async = true; gcse.src = 'https://cse.google.com/cse.js?cx=' + cx; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(gcse, s); })(); </script> <meta http-equiv="keywords" name="keywords" content="product rule, quotient rule" /><meta http-equiv="description" name="description" content="In this section we will give two of the more important formulas for differentiating functions. We will discuss the Product Rule and the Quotient Rule allowing us to differentiate functions that, up to this point, we were unable to differentiate." /></head> <body onload="init({Notes: 'NoteMobile;8/21/2018;true'})"> <div id="page"> <div class="header"> <div class="header-row">  <div id="side-menu-icon" class="header-side-menu-icon"><a href="#menu"><span class="fas fa-bars fa-lg" aria-hidden="true" title="Main Menu - Change between topics, chapters and sections as well as a few extra pages."></span></a></div> <span class="header-title"><a href="/" class="header-title-link">Paul's Online Notes</a></span> <div class="header-spacer"></div> <div id="content-top-menu" class="top-menu"> <button id="content-type-menu" class="top-menu-button" data-jq-dropdown="#jq-dropdown-type" title="View (Notes, Practice Problems or Assignment Problems, Show/Hide Solutions and/or Steps) Menu"> <span id="tab_top_menu_notes" class="top-menu-item-title">Notes</span> <span class="far fa-eye fa-lg" aria-hidden="true"></span> </button> <button id="quicknav-menu" class="top-menu-button" data-jq-dropdown="#jq-dropdown-quicknav" title="Quick Navigation (Previous/Next Sections and Problems and Full Problem List) Menu"> <span class="top-menu-item-title">Quick Nav</span> <span class="fas fa-exchange fa-lg" aria-hidden="true"></span> </button> <button id="download-menu" class="top-menu-button" data-jq-dropdown="#jq-dropdown-download" title="Download pdf Menu"> <span class="top-menu-item-title">Download</span> <span class="far fa-download fa-lg" aria-hidden="true"></span> </button> <button id="print-menu" class="top-menu-button top-menu-button-icon-only" data-jq-dropdown="#jq-print-download" title="Print Menu"> <span class="far fa-print fa-lg" aria-hidden="true"></span> </button> </div> <div id="header-google-search" class="header-search"> <gcse:search></gcse:search> </div> <div id="header-search-icon" title="Site Search" class="header-menu-icon"><span id="search-icon" class="fas fa-search" aria-hidden="true"></span></div> </div> </div> <div id="jq-dropdown-type" class="jq-dropdown jq-dropdown-tip"> <ul class="jq-dropdown-menu"> <li id="li_type_menu_goto" class="top-menu-nav-title">Go To</li> <li id="li_type_menu_notes"> <span id="type_menu_notes_span" title="Viewing the Notes for the current topic." class="top-menu-current">Notes</span> </li> <li id="li_type_menu_practice"> <a href="/Problems/CalcI/ProductQuotientRule.aspx" id="type_menu_practice_link" title="Go to Practice Problems for current topic.">Practice Problems</a> </li> <li id="li_type_menu_asgn"> <a href="/ProblemsNS/CalcI/ProductQuotientRule.aspx" id="type_menu_asgn_link" title="Go to Assignment Problems for current topic.">Assignment Problems</a> </li> <li id="li_type_menu_sh" class="top-menu-nav-title">Show/Hide</li> <li id="li_type_menu_show" title="Show any hidden solutions and/or steps that may be present on the page."><a href="javascript:SHPrintPage(1,0)" id="view_menu_show">Show all Solutions/Steps/<em>etc.</em></a></li> <li id="li_type_menu_hide" title="Hide any visible solutions and/or steps that may be present on the page."><a href="javascript:SHPrintPage(0,0)" id="view_menu_hide">Hide all Solutions/Steps/<em>etc.</em></a></li> </ul> </div> <div id="jq-dropdown-quicknav" class="jq-dropdown jq-dropdown-tip"> <ul class="jq-dropdown-menu"> <li id="li_nav_menu_sections" class="top-menu-nav-title">Sections</li> <li id="li_nav_menu_prev_section"><a href="/Classes/CalcI/DiffFormulas.aspx" id="a_nav_menu_prev_section" class="top-menu-nav-link" title="Previous Section : Differentiation Formulas"><span class="top-menu-prev fas fa-chevron-left"></span> Differentiation Formulas</a></li> <li id="li_nav_menu_next_section"><a href="/Classes/CalcI/DiffTrigFcns.aspx" id="a_nav_menu_next_section" class="top-menu-nav-link" title="Next Section : Derivatives of Trig Functions"><span class="top-menu-prev-hidden fas fa-chevron-left"></span> Derivatives of Trig Functions <span class="top-menu-next fas fa-chevron-right"></span></a></li> <li id="li_nav_menu_chapters" class="top-menu-nav-title">Chapters</li> <li id="li_nav_menu_prev_chapter"><a href="/Classes/CalcI/limitsIntro.aspx" id="a_nav_menu_prev_chapter" class="top-menu-nav-link" title="Previous Chapter : Limits"><span class="top-menu-prev fas fa-chevron-left"></span><span class="top-menu-prev fas fa-chevron-left"></span> Limits</a></li> <li id="li_nav_menu_next_chapter"><a href="/Classes/CalcI/DerivAppsIntro.aspx" id="a_nav_menu_next_chapter" class="top-menu-nav-link" title="Next Chapter : Applications of Derivatives"><span class="top-menu-prev-hidden fas fa-chevron-left"></span><span class="top-menu-prev-hidden fas fa-chevron-left"></span> Applications of Derivatives <span class="top-menu-next fas fa-chevron-right"></span><span class="top-menu-next fas fa-chevron-right"></span></a></li> <li id="li_nav_menu_classes" class="top-menu-nav-title">Classes</li> <li> <a href="/Classes/Alg/Alg.aspx" id="nav_menu_alg_link" title="Go To Algebra Notes">Algebra</a> </li> <li> <span id="nav_menu_calci_span" title="Currently Viewing Calculus I Material" class="top-menu-current">Calculus I</span> </li> <li> <a href="/Classes/CalcII/CalcII.aspx" id="nav_menu_calcii_link" title="Go To Calculus II Notes">Calculus II</a> </li> <li> <a href="/Classes/CalcIII/CalcIII.aspx" id="nav_menu_calciii_link" title="Go To Calculus III Notes">Calculus III</a> </li> <li> <a href="/Classes/DE/DE.aspx" id="nav_menu_de_link" title="Go To Differential Equations Notes">Differential Equations</a> </li> <li id="li_nav_menu_extras" class="top-menu-nav-title">Extras</li> <li> <a href="/Extras/AlgebraTrigReview/AlgebraTrig.aspx" id="nav_menu_algtrig_link" title="Go To Algebra & Trig Review">Algebra & Trig Review</a> </li> <li> <a href="/Extras/CommonErrors/CommonMathErrors.aspx" id="nav_menu_commonerrors_link" title="Go To Common Math Errors">Common Math Errors</a> </li> <li> <a href="/Extras/ComplexPrimer/ComplexNumbers.aspx" id="nav_menu_complexnumbers_link" title="Go To Complex Numbers Primer">Complex Number Primer</a> </li> <li> <a href="/Extras/StudyMath/HowToStudyMath.aspx" id="nav_menu_studymath_link" title="Go To How To Study Math">How To Study Math</a> </li> <li> <a href="/Extras/CheatSheets_Tables.aspx" id="nav_menu_cheattables_link" title="Go To List of Cheat Sheets and Tables">Cheat Sheets & Tables</a> </li> <li id="li_nav_menu_misc" class="top-menu-nav-title">Misc</li> <li><a href="/contact.aspx" id="nav_menu_contact" title="Contact Me!">Contact Me</a></li> <li><a href="/mathjax.aspx" id="nav_menu_mathjax" title="Info on MathJax and MathJax Configuration Menu">MathJax Help and Configuration</a></li> </ul> </div> <div id="jq-dropdown-download" class="jq-dropdown jq-dropdown-anchor-right jq-dropdown-tip"> <ul class="jq-dropdown-menu"> <li id="li_download_menu_notes" class="top-menu-nav-title">Notes Downloads</li> <li id="li_download_menu_notes_book"><a href="/GetFile.aspx?file=B,20,N" id="download_menu_notes_book" data-PDF="Download - Menu$Notes - Book$Calculus$/Downloads/Calculus/Notes/Complete.pdf">Complete Book</a></li> <li id="li_download_menu_practice" class="top-menu-nav-title">Practice Problems Downloads</li> <li id="li_download_menu_practice_book"><a href="/GetFile.aspx?file=B,20,P" id="download_menu_practice_book" data-PDF="Download - Menu$Practice Problems - Book$Calculus$/Downloads/Calculus/Practice Problems/Complete.pdf">Complete Book - Problems Only</a></li> <li id="li_download_menu_solutions_book"><a href="/GetFile.aspx?file=B,20,S" id="download_menu_solutions_book" data-PDF="Download - Menu$Practice Problems Solutions - Book$Calculus$/Downloads/Calculus/Practice Problems Solutions/Complete.pdf">Complete Book - Solutions</a></li> <li id="li_download_menu_asgn" class="top-menu-nav-title">Assignment Problems Downloads</li> <li id="li_download_menu_asgn_book"><a href="/GetFile.aspx?file=B,20,A" id="download_menu_asgn_book" data-PDF="Download - Menu$Assignment Problems - Book$Calculus$/Downloads/Calculus/Assignment Problems/Complete.pdf">Complete Book</a></li> <li id="li_download_menu_other" class="top-menu-nav-title">Other Items</li> <li id="li_download_menu_urls"> <a href="/DownloadURLs.aspx?bi=20" id="download_menu_urls">Get URL's for Download Items</a> </li> </ul> </div> <div id="jq-print-download" class="jq-dropdown jq-dropdown-anchor-right jq-dropdown-tip"> <ul class="jq-dropdown-menu"> <li id="li_print_menu_default"><a href="javascript:SHPrintPage()" id="print_menu_default">Print Page in Current Form (Default)</a></li> <li id="li_print_menu_show"><a href="javascript:SHPrintPage(1,1)" id="print_menu_show">Show all Solutions/Steps and Print Page</a></li> <li id="li_print_menu_hide"><a href="javascript:SHPrintPage(0,1)" id="print_menu_hide">Hide all Solutions/Steps and Print Page</a></li> </ul> </div> <nav id="menu" class="notes-nav"> <ul> <li><a href="/" class="mm-link">Home</a></li> <li><span>Classes</span></li> <li><a href="/Classes/Alg/Alg.aspx" class="mm-link">Algebra</a> <ul> <li><a href="/Classes/Alg/Preliminaries.aspx" class="mm-link">1. Preliminaries</a> <ul> <li><a href="/Classes/Alg/IntegerExponents.aspx" class="mm-link">1.1 Integer Exponents</a></li> <li><a href="/Classes/Alg/RationalExponents.aspx" class="mm-link">1.2 Rational Exponents</a></li> <li><a href="/Classes/Alg/Radicals.aspx" class="mm-link">1.3 Radicals</a></li> <li><a href="/Classes/Alg/Polynomials.aspx" class="mm-link">1.4 Polynomials</a></li> <li><a href="/Classes/Alg/Factoring.aspx" class="mm-link">1.5 Factoring Polynomials</a></li> <li><a href="/Classes/Alg/RationalExpressions.aspx" class="mm-link">1.6 Rational Expressions</a></li> <li><a href="/Classes/Alg/ComplexNumbers.aspx" class="mm-link">1.7 Complex Numbers</a></li> </ul> </li> <li><a href="/Classes/Alg/Solving.aspx" class="mm-link">2. Solving Equations and Inequalities</a> <ul> <li><a href="/Classes/Alg/SolutionSets.aspx" class="mm-link">2.1 Solutions and Solution Sets</a></li> <li><a href="/Classes/Alg/SolveLinearEqns.aspx" class="mm-link">2.2 Linear Equations</a></li> <li><a href="/Classes/Alg/LinearApps.aspx" class="mm-link">2.3 Applications of Linear Equations</a></li> <li><a href="/Classes/Alg/SolveMultiVariable.aspx" class="mm-link">2.4 Equations With More Than One Variable</a></li> <li><a href="/Classes/Alg/SolveQuadraticEqnsI.aspx" class="mm-link">2.5 Quadratic Equations - Part I</a></li> <li><a href="/Classes/Alg/SolveQuadraticEqnsII.aspx" class="mm-link">2.6 Quadratic Equations - Part II</a></li> <li><a href="/Classes/Alg/SolveQuadraticEqnSummary.aspx" class="mm-link">2.7 Quadratic Equations : A Summary</a></li> <li><a href="/Classes/Alg/QuadraticApps.aspx" class="mm-link">2.8 Applications of Quadratic Equations</a></li> <li><a href="/Classes/Alg/ReducibleToQuadratic.aspx" class="mm-link">2.9 Equations Reducible to Quadratic in Form</a></li> <li><a href="/Classes/Alg/SolveRadicalEqns.aspx" class="mm-link">2.10 Equations with Radicals</a></li> <li><a href="/Classes/Alg/SolveLinearInequalities.aspx" class="mm-link">2.11 Linear Inequalities</a></li> <li><a href="/Classes/Alg/SolvePolyInequalities.aspx" class="mm-link">2.12 Polynomial Inequalities</a></li> <li><a href="/Classes/Alg/SolveRationalInequalities.aspx" class="mm-link">2.13 Rational Inequalities</a></li> <li><a href="/Classes/Alg/SolveAbsValueEqns.aspx" class="mm-link">2.14 Absolute Value Equations</a></li> <li><a href="/Classes/Alg/SolveAbsValueIneq.aspx" class="mm-link">2.15 Absolute Value Inequalities</a></li> </ul> </li> <li><a href="/Classes/Alg/Graphing_Functions.aspx" class="mm-link">3. Graphing and Functions</a> <ul> <li><a href="/Classes/Alg/Graphing.aspx" class="mm-link">3.1 Graphing</a></li> <li><a href="/Classes/Alg/Lines.aspx" class="mm-link">3.2 Lines</a></li> <li><a href="/Classes/Alg/Circles.aspx" class="mm-link">3.3 Circles</a></li> <li><a href="/Classes/Alg/FunctionDefn.aspx" class="mm-link">3.4 The Definition of a Function</a></li> <li><a href="/Classes/Alg/GraphFunctions.aspx" class="mm-link">3.5 Graphing Functions</a></li> <li><a href="/Classes/Alg/CombineFunctions.aspx" class="mm-link">3.6 Combining Functions</a></li> <li><a href="/Classes/Alg/InverseFunctions.aspx" class="mm-link">3.7 Inverse Functions</a></li> </ul> </li> <li><a href="/Classes/Alg/CommonGraphs.aspx" class="mm-link">4. Common Graphs</a> <ul> <li><a href="/Classes/Alg/Lines_Circles_PWF.aspx" class="mm-link">4.1 Lines, Circles and Piecewise Functions</a></li> <li><a href="/Classes/Alg/Parabolas.aspx" class="mm-link">4.2 Parabolas</a></li> <li><a href="/Classes/Alg/Ellipses.aspx" class="mm-link">4.3 Ellipses</a></li> <li><a href="/Classes/Alg/Hyperbolas.aspx" class="mm-link">4.4 Hyperbolas</a></li> <li><a href="/Classes/Alg/MiscFunctions.aspx" class="mm-link">4.5 Miscellaneous Functions</a></li> <li><a href="/Classes/Alg/Transformations.aspx" class="mm-link">4.6 Transformations</a></li> <li><a href="/Classes/Alg/Symmetry.aspx" class="mm-link">4.7 Symmetry</a></li> <li><a href="/Classes/Alg/GraphRationalFcns.aspx" class="mm-link">4.8 Rational Functions</a></li> </ul> </li> <li><a href="/Classes/Alg/PolynomialFunctions.aspx" class="mm-link">5. Polynomial Functions</a> <ul> <li><a href="/Classes/Alg/DividingPolynomials.aspx" class="mm-link">5.1 Dividing Polynomials</a></li> <li><a href="/Classes/Alg/ZeroesOfPolynomials.aspx" class="mm-link">5.2 Zeroes/Roots of Polynomials</a></li> <li><a href="/Classes/Alg/GraphingPolynomials.aspx" class="mm-link">5.3 Graphing Polynomials</a></li> <li><a href="/Classes/Alg/FindingZeroesOfPolynomials.aspx" class="mm-link">5.4 Finding Zeroes of Polynomials</a></li> <li><a href="/Classes/Alg/PartialFractions.aspx" class="mm-link">5.5 Partial Fractions</a></li> </ul> </li> <li><a href="/Classes/Alg/ExpAndLog.aspx" class="mm-link">6. Exponential and Logarithm Functions</a> <ul> <li><a href="/Classes/Alg/ExpFunctions.aspx" class="mm-link">6.1 Exponential Functions</a></li> <li><a href="/Classes/Alg/LogFunctions.aspx" class="mm-link">6.2 Logarithm Functions</a></li> <li><a href="/Classes/Alg/SolveExpEqns.aspx" class="mm-link">6.3 Solving Exponential Equations</a></li> <li><a href="/Classes/Alg/SolveLogEqns.aspx" class="mm-link">6.4 Solving Logarithm Equations</a></li> <li><a href="/Classes/Alg/ExpLogApplications.aspx" class="mm-link">6.5 Applications</a></li> </ul> </li> <li><a href="/Classes/Alg/Systems.aspx" class="mm-link">7. Systems of Equations</a> <ul> <li><a href="/Classes/Alg/SystemsTwoVrble.aspx" class="mm-link">7.1 Linear Systems with Two Variables</a></li> <li><a href="/Classes/Alg/SystemsThreeVrble.aspx" class="mm-link">7.2 Linear Systems with Three Variables</a></li> <li><a href="/Classes/Alg/AugmentedMatrix.aspx" class="mm-link">7.3 Augmented Matrices</a></li> <li><a href="/Classes/Alg/AugmentedMatrixII.aspx" class="mm-link">7.4 More on the Augmented Matrix</a></li> <li><a href="/Classes/Alg/NonlinearSystems.aspx" class="mm-link">7.5 Nonlinear Systems</a></li> </ul> </li> </ul> </li> <li><a href="/Classes/CalcI/CalcI.aspx" class="mm-link">Calculus I</a> <ul> <li><a href="/Classes/CalcI/ReviewIntro.aspx" class="mm-link">1. Review</a> <ul> <li><a href="/Classes/CalcI/Functions.aspx" class="mm-link">1.1 Functions</a></li> <li><a href="/Classes/CalcI/InverseFunctions.aspx" class="mm-link">1.2 Inverse Functions</a></li> <li><a href="/Classes/CalcI/TrigFcns.aspx" class="mm-link">1.3 Trig Functions</a></li> <li><a href="/Classes/CalcI/TrigEquations.aspx" class="mm-link">1.4 Solving Trig Equations</a></li> <li><a href="/Classes/CalcI/TrigEquations_CalcI.aspx" class="mm-link">1.5 Trig Equations with Calculators, Part I</a></li> <li><a href="/Classes/CalcI/TrigEquations_CalcII.aspx" class="mm-link">1.6 Trig Equations with Calculators, Part II</a></li> <li><a href="/Classes/CalcI/ExpFunctions.aspx" class="mm-link">1.7 Exponential Functions</a></li> <li><a href="/Classes/CalcI/LogFcns.aspx" class="mm-link">1.8 Logarithm Functions</a></li> <li><a href="/Classes/CalcI/ExpLogEqns.aspx" class="mm-link">1.9 Exponential and Logarithm Equations</a></li> <li><a href="/Classes/CalcI/CommonGraphs.aspx" class="mm-link">1.10 Common Graphs</a></li> </ul> </li> <li><a href="/Classes/CalcI/limitsIntro.aspx" class="mm-link">2. Limits</a> <ul> <li><a href="/Classes/CalcI/Tangents_Rates.aspx" class="mm-link">2.1 Tangent Lines and Rates of Change</a></li> <li><a href="/Classes/CalcI/TheLimit.aspx" class="mm-link">2.2 The Limit</a></li> <li><a href="/Classes/CalcI/OneSidedLimits.aspx" class="mm-link">2.3 One-Sided Limits</a></li> <li><a href="/Classes/CalcI/LimitsProperties.aspx" class="mm-link">2.4 Limit Properties</a></li> <li><a href="/Classes/CalcI/ComputingLimits.aspx" class="mm-link">2.5 Computing Limits</a></li> <li><a href="/Classes/CalcI/InfiniteLimits.aspx" class="mm-link">2.6 Infinite Limits</a></li> <li><a href="/Classes/CalcI/LimitsAtInfinityI.aspx" class="mm-link">2.7 Limits At Infinity, Part I</a></li> <li><a href="/Classes/CalcI/LimitsAtInfinityII.aspx" class="mm-link">2.8 Limits At Infinity, Part II</a></li> <li><a href="/Classes/CalcI/Continuity.aspx" class="mm-link">2.9 Continuity</a></li> <li><a href="/Classes/CalcI/DefnOfLimit.aspx" class="mm-link">2.10 The Definition of the Limit</a></li> </ul> </li> <li><a href="/Classes/CalcI/DerivativeIntro.aspx" class="mm-link">3. Derivatives</a> <ul> <li><a href="/Classes/CalcI/DefnOfDerivative.aspx" class="mm-link">3.1 The Definition of the Derivative</a></li> <li><a href="/Classes/CalcI/DerivativeInterp.aspx" class="mm-link">3.2 Interpretation of the Derivative</a></li> <li><a href="/Classes/CalcI/DiffFormulas.aspx" class="mm-link">3.3 Differentiation Formulas</a></li> <li><a href="/Classes/CalcI/ProductQuotientRule.aspx" class="mm-link">3.4 Product and Quotient Rule</a></li> <li><a href="/Classes/CalcI/DiffTrigFcns.aspx" class="mm-link">3.5 Derivatives of Trig Functions</a></li> <li><a href="/Classes/CalcI/DiffExpLogFcns.aspx" class="mm-link">3.6 Derivatives of Exponential and Logarithm Functions</a></li> <li><a href="/Classes/CalcI/DiffInvTrigFcns.aspx" class="mm-link">3.7 Derivatives of Inverse Trig Functions</a></li> <li><a href="/Classes/CalcI/DiffHyperFcns.aspx" class="mm-link">3.8 Derivatives of Hyperbolic Functions</a></li> <li><a href="/Classes/CalcI/ChainRule.aspx" class="mm-link">3.9 Chain Rule</a></li> <li><a href="/Classes/CalcI/ImplicitDIff.aspx" class="mm-link">3.10 Implicit Differentiation</a></li> <li><a href="/Classes/CalcI/RelatedRates.aspx" class="mm-link">3.11 Related Rates</a></li> <li><a href="/Classes/CalcI/HigherOrderDerivatives.aspx" class="mm-link">3.12 Higher Order Derivatives</a></li> <li><a href="/Classes/CalcI/LogDiff.aspx" class="mm-link">3.13 Logarithmic Differentiation</a></li> </ul> </li> <li><a href="/Classes/CalcI/DerivAppsIntro.aspx" class="mm-link">4. Applications of Derivatives</a> <ul> <li><a href="/Classes/CalcI/RateOfChange.aspx" class="mm-link">4.1 Rates of Change</a></li> <li><a href="/Classes/CalcI/CriticalPoints.aspx" class="mm-link">4.2 Critical Points</a></li> <li><a href="/Classes/CalcI/MinMaxValues.aspx" class="mm-link">4.3 Minimum and Maximum Values</a></li> <li><a href="/Classes/CalcI/AbsExtrema.aspx" class="mm-link">4.4 Finding Absolute Extrema</a></li> <li><a href="/Classes/CalcI/ShapeofGraphPtI.aspx" class="mm-link">4.5 The Shape of a Graph, Part I</a></li> <li><a href="/Classes/CalcI/ShapeofGraphPtII.aspx" class="mm-link">4.6 The Shape of a Graph, Part II</a></li> <li><a href="/Classes/CalcI/MeanValueTheorem.aspx" class="mm-link">4.7 The Mean Value Theorem</a></li> <li><a href="/Classes/CalcI/Optimization.aspx" class="mm-link">4.8 Optimization</a></li> <li><a href="/Classes/CalcI/MoreOptimization.aspx" class="mm-link">4.9 More Optimization Problems</a></li> <li><a href="/Classes/CalcI/LHospitalsRule.aspx" class="mm-link">4.10 L'Hospital's Rule and Indeterminate Forms</a></li> <li><a href="/Classes/CalcI/LinearApproximations.aspx" class="mm-link">4.11 Linear Approximations</a></li> <li><a href="/Classes/CalcI/Differentials.aspx" class="mm-link">4.12 Differentials</a></li> <li><a href="/Classes/CalcI/NewtonsMethod.aspx" class="mm-link">4.13 Newton's Method</a></li> <li><a href="/Classes/CalcI/BusinessApps.aspx" class="mm-link">4.14 Business Applications</a></li> </ul> </li> <li><a href="/Classes/CalcI/IntegralsIntro.aspx" class="mm-link">5. Integrals</a> <ul> <li><a href="/Classes/CalcI/IndefiniteIntegrals.aspx" class="mm-link">5.1 Indefinite Integrals</a></li> <li><a href="/Classes/CalcI/ComputingIndefiniteIntegrals.aspx" class="mm-link">5.2 Computing Indefinite Integrals</a></li> <li><a href="/Classes/CalcI/SubstitutionRuleIndefinite.aspx" class="mm-link">5.3 Substitution Rule for Indefinite Integrals</a></li> <li><a href="/Classes/CalcI/SubstitutionRuleIndefinitePtII.aspx" class="mm-link">5.4 More Substitution Rule</a></li> <li><a href="/Classes/CalcI/AreaProblem.aspx" class="mm-link">5.5 Area Problem</a></li> <li><a href="/Classes/CalcI/DefnOfDefiniteIntegral.aspx" class="mm-link">5.6 Definition of the Definite Integral</a></li> <li><a href="/Classes/CalcI/ComputingDefiniteIntegrals.aspx" class="mm-link">5.7 Computing Definite Integrals</a></li> <li><a href="/Classes/CalcI/SubstitutionRuleDefinite.aspx" class="mm-link">5.8 Substitution Rule for Definite Integrals</a></li> </ul> </li> <li><a href="/Classes/CalcI/IntAppsIntro.aspx" class="mm-link">6. Applications of Integrals</a> <ul> <li><a href="/Classes/CalcI/AvgFcnValue.aspx" class="mm-link">6.1 Average Function Value</a></li> <li><a href="/Classes/CalcI/AreaBetweenCurves.aspx" class="mm-link">6.2 Area Between Curves</a></li> <li><a href="/Classes/CalcI/VolumeWithRings.aspx" class="mm-link">6.3 Volumes of Solids of Revolution / Method of Rings</a></li> <li><a href="/Classes/CalcI/VolumeWithCylinder.aspx" class="mm-link">6.4 Volumes of Solids of Revolution/Method of Cylinders</a></li> <li><a href="/Classes/CalcI/MoreVolume.aspx" class="mm-link">6.5 More Volume Problems</a></li> <li><a href="/Classes/CalcI/Work.aspx" class="mm-link">6.6 Work</a></li> </ul> </li> <li><a href="/Classes/CalcI/ExtrasIntro.aspx" class="mm-link">Appendix A. Extras</a> <ul> <li><a href="/Classes/CalcI/LimitProofs.aspx" class="mm-link">A.1 Proof of Various Limit Properties</a></li> <li><a href="/Classes/CalcI/DerivativeProofs.aspx" class="mm-link">A.2 Proof of Various Derivative Properties</a></li> <li><a href="/Classes/CalcI/ProofTrigDeriv.aspx" class="mm-link">A.3 Proof of Trig Limits</a></li> <li><a href="/Classes/CalcI/DerivativeAppsProofs.aspx" class="mm-link">A.4 Proofs of Derivative Applications Facts</a></li> <li><a href="/Classes/CalcI/ProofIntProp.aspx" class="mm-link">A.5 Proof of Various Integral Properties </a></li> <li><a href="/Classes/CalcI/Area_Volume_Formulas.aspx" class="mm-link">A.6 Area and Volume Formulas</a></li> <li><a href="/Classes/CalcI/TypesOfInfinity.aspx" class="mm-link">A.7 Types of Infinity</a></li> <li><a href="/Classes/CalcI/SummationNotation.aspx" class="mm-link">A.8 Summation Notation</a></li> <li><a href="/Classes/CalcI/ConstantofIntegration.aspx" class="mm-link">A.9 Constant of Integration</a></li> </ul> </li> </ul> </li> <li><a href="/Classes/CalcII/CalcII.aspx" class="mm-link">Calculus II</a> <ul> <li><a href="/Classes/CalcII/IntTechIntro.aspx" class="mm-link">7. Integration Techniques</a> <ul> <li><a href="/Classes/CalcII/IntegrationByParts.aspx" class="mm-link">7.1 Integration by Parts</a></li> <li><a href="/Classes/CalcII/IntegralsWithTrig.aspx" class="mm-link">7.2 Integrals Involving Trig Functions</a></li> <li><a href="/Classes/CalcII/TrigSubstitutions.aspx" class="mm-link">7.3 Trig Substitutions</a></li> <li><a href="/Classes/CalcII/PartialFractions.aspx" class="mm-link">7.4 Partial Fractions</a></li> <li><a href="/Classes/CalcII/IntegralsWithRoots.aspx" class="mm-link">7.5 Integrals Involving Roots</a></li> <li><a href="/Classes/CalcII/IntegralsWithQuadratics.aspx" class="mm-link">7.6 Integrals Involving Quadratics</a></li> <li><a href="/Classes/CalcII/IntegrationStrategy.aspx" class="mm-link">7.7 Integration Strategy</a></li> <li><a href="/Classes/CalcII/ImproperIntegrals.aspx" class="mm-link">7.8 Improper Integrals</a></li> <li><a href="/Classes/CalcII/ImproperIntegralsCompTest.aspx" class="mm-link">7.9 Comparison Test for Improper Integrals</a></li> <li><a href="/Classes/CalcII/ApproximatingDefIntegrals.aspx" class="mm-link">7.10 Approximating Definite Integrals</a></li> </ul> </li> <li><a href="/Classes/CalcII/IntAppsIntro.aspx" class="mm-link">8. Applications of Integrals</a> <ul> <li><a href="/Classes/CalcII/ArcLength.aspx" class="mm-link">8.1 Arc Length</a></li> <li><a href="/Classes/CalcII/SurfaceArea.aspx" class="mm-link">8.2 Surface Area</a></li> <li><a href="/Classes/CalcII/CenterOfMass.aspx" class="mm-link">8.3 Center of Mass</a></li> <li><a href="/Classes/CalcII/HydrostaticPressure.aspx" class="mm-link">8.4 Hydrostatic Pressure</a></li> <li><a href="/Classes/CalcII/Probability.aspx" class="mm-link">8.5 Probability</a></li> </ul> </li> <li><a href="/Classes/CalcII/ParametricIntro.aspx" class="mm-link">9. Parametric Equations and Polar Coordinates</a> <ul> <li><a href="/Classes/CalcII/ParametricEqn.aspx" class="mm-link">9.1 Parametric Equations and Curves</a></li> <li><a href="/Classes/CalcII/ParaTangent.aspx" class="mm-link">9.2 Tangents with Parametric Equations</a></li> <li><a href="/Classes/CalcII/ParaArea.aspx" class="mm-link">9.3 Area with Parametric Equations</a></li> <li><a href="/Classes/CalcII/ParaArcLength.aspx" class="mm-link">9.4 Arc Length with Parametric Equations</a></li> <li><a href="/Classes/CalcII/ParaSurfaceArea.aspx" class="mm-link">9.5 Surface Area with Parametric Equations</a></li> <li><a href="/Classes/CalcII/PolarCoordinates.aspx" class="mm-link">9.6 Polar Coordinates</a></li> <li><a href="/Classes/CalcII/PolarTangents.aspx" class="mm-link">9.7 Tangents with Polar Coordinates</a></li> <li><a href="/Classes/CalcII/PolarArea.aspx" class="mm-link">9.8 Area with Polar Coordinates</a></li> <li><a href="/Classes/CalcII/PolarArcLength.aspx" class="mm-link">9.9 Arc Length with Polar Coordinates</a></li> <li><a href="/Classes/CalcII/PolarSurfaceArea.aspx" class="mm-link">9.10 Surface Area with Polar Coordinates</a></li> <li><a href="/Classes/CalcII/ArcLength_SurfaceArea.aspx" class="mm-link">9.11 Arc Length and Surface Area Revisited</a></li> </ul> </li> <li><a href="/Classes/CalcII/SeriesIntro.aspx" class="mm-link">10. Series & Sequences</a> <ul> <li><a href="/Classes/CalcII/Sequences.aspx" class="mm-link">10.1 Sequences</a></li> <li><a href="/Classes/CalcII/MoreSequences.aspx" class="mm-link">10.2 More on Sequences</a></li> <li><a href="/Classes/CalcII/Series_Basics.aspx" class="mm-link">10.3 Series - The Basics</a></li> <li><a href="/Classes/CalcII/ConvergenceOfSeries.aspx" class="mm-link">10.4 Convergence/Divergence of Series</a></li> <li><a href="/Classes/CalcII/Series_Special.aspx" class="mm-link">10.5 Special Series</a></li> <li><a href="/Classes/CalcII/IntegralTest.aspx" class="mm-link">10.6 Integral Test</a></li> <li><a href="/Classes/CalcII/SeriesCompTest.aspx" class="mm-link">10.7 Comparison Test/Limit Comparison Test</a></li> <li><a href="/Classes/CalcII/AlternatingSeries.aspx" class="mm-link">10.8 Alternating Series Test</a></li> <li><a href="/Classes/CalcII/AbsoluteConvergence.aspx" class="mm-link">10.9 Absolute Convergence</a></li> <li><a href="/Classes/CalcII/RatioTest.aspx" class="mm-link">10.10 Ratio Test</a></li> <li><a href="/Classes/CalcII/RootTest.aspx" class="mm-link">10.11 Root Test</a></li> <li><a href="/Classes/CalcII/SeriesStrategy.aspx" class="mm-link">10.12 Strategy for Series</a></li> <li><a href="/Classes/CalcII/EstimatingSeries.aspx" class="mm-link">10.13 Estimating the Value of a Series</a></li> <li><a href="/Classes/CalcII/PowerSeries.aspx" class="mm-link">10.14 Power Series</a></li> <li><a href="/Classes/CalcII/PowerSeriesandFunctions.aspx" class="mm-link">10.15 Power Series and Functions</a></li> <li><a href="/Classes/CalcII/TaylorSeries.aspx" class="mm-link">10.16 Taylor Series</a></li> <li><a href="/Classes/CalcII/TaylorSeriesApps.aspx" class="mm-link">10.17 Applications of Series</a></li> <li><a href="/Classes/CalcII/BinomialSeries.aspx" class="mm-link">10.18 Binomial Series</a></li> </ul> </li> <li><a href="/Classes/CalcII/VectorsIntro.aspx" class="mm-link">11. Vectors</a> <ul> <li><a href="/Classes/CalcII/Vectors_Basics.aspx" class="mm-link">11.1 Vectors - The Basics</a></li> <li><a href="/Classes/CalcII/VectorArithmetic.aspx" class="mm-link">11.2 Vector Arithmetic</a></li> <li><a href="/Classes/CalcII/DotProduct.aspx" class="mm-link">11.3 Dot Product</a></li> <li><a href="/Classes/CalcII/CrossProduct.aspx" class="mm-link">11.4 Cross Product</a></li> </ul> </li> <li><a href="/Classes/CalcII/3DSpace.aspx" class="mm-link">12. 3-Dimensional Space</a> <ul> <li><a href="/Classes/CalcII/3DCoords.aspx" class="mm-link">12.1 The 3-D Coordinate System</a></li> <li><a href="/Classes/CalcII/EqnsOfLines.aspx" class="mm-link">12.2 Equations of Lines</a></li> <li><a href="/Classes/CalcII/EqnsOfPlanes.aspx" class="mm-link">12.3 Equations of Planes</a></li> <li><a href="/Classes/CalcII/QuadricSurfaces.aspx" class="mm-link">12.4 Quadric Surfaces</a></li> <li><a href="/Classes/CalcII/MultiVrbleFcns.aspx" class="mm-link">12.5 Functions of Several Variables</a></li> <li><a href="/Classes/CalcII/VectorFunctions.aspx" class="mm-link">12.6 Vector Functions</a></li> <li><a href="/Classes/CalcII/VectorFcnsCalculus.aspx" class="mm-link">12.7 Calculus with Vector Functions</a></li> <li><a href="/Classes/CalcII/TangentNormalVectors.aspx" class="mm-link">12.8 Tangent, Normal and Binormal Vectors</a></li> <li><a href="/Classes/CalcII/VectorArcLength.aspx" class="mm-link">12.9 Arc Length with Vector Functions</a></li> <li><a href="/Classes/CalcII/Curvature.aspx" class="mm-link">12.10 Curvature</a></li> <li><a href="/Classes/CalcII/Velocity_Acceleration.aspx" class="mm-link">12.11 Velocity and Acceleration</a></li> <li><a href="/Classes/CalcII/CylindricalCoords.aspx" class="mm-link">12.12 Cylindrical Coordinates</a></li> <li><a href="/Classes/CalcII/SphericalCoords.aspx" class="mm-link">12.13 Spherical Coordinates</a></li> </ul> </li> </ul> </li> <li><a href="/Classes/CalcIII/CalcIII.aspx" class="mm-link">Calculus III</a> <ul> <li><a href="/Classes/CalcIII/3DSpace.aspx" class="mm-link">12. 3-Dimensional Space</a> <ul> <li><a href="/Classes/CalcIII/3DCoords.aspx" class="mm-link">12.1 The 3-D Coordinate System</a></li> <li><a href="/Classes/CalcIII/EqnsOfLines.aspx" class="mm-link">12.2 Equations of Lines</a></li> <li><a href="/Classes/CalcIII/EqnsOfPlanes.aspx" class="mm-link">12.3 Equations of Planes</a></li> <li><a href="/Classes/CalcIII/QuadricSurfaces.aspx" class="mm-link">12.4 Quadric Surfaces</a></li> <li><a href="/Classes/CalcIII/MultiVrbleFcns.aspx" class="mm-link">12.5 Functions of Several Variables</a></li> <li><a href="/Classes/CalcIII/VectorFunctions.aspx" class="mm-link">12.6 Vector Functions</a></li> <li><a href="/Classes/CalcIII/VectorFcnsCalculus.aspx" class="mm-link">12.7 Calculus with Vector Functions</a></li> <li><a href="/Classes/CalcIII/TangentNormalVectors.aspx" class="mm-link">12.8 Tangent, Normal and Binormal Vectors</a></li> <li><a href="/Classes/CalcIII/VectorArcLength.aspx" class="mm-link">12.9 Arc Length with Vector Functions</a></li> <li><a href="/Classes/CalcIII/Curvature.aspx" class="mm-link">12.10 Curvature</a></li> <li><a href="/Classes/CalcIII/Velocity_Acceleration.aspx" class="mm-link">12.11 Velocity and Acceleration</a></li> <li><a href="/Classes/CalcIII/CylindricalCoords.aspx" class="mm-link">12.12 Cylindrical Coordinates</a></li> <li><a href="/Classes/CalcIII/SphericalCoords.aspx" class="mm-link">12.13 Spherical Coordinates</a></li> </ul> </li> <li><a href="/Classes/CalcIII/PartialDerivsIntro.aspx" class="mm-link">13. Partial Derivatives</a> <ul> <li><a href="/Classes/CalcIII/Limits.aspx" class="mm-link">13.1 Limits</a></li> <li><a href="/Classes/CalcIII/PartialDerivatives.aspx" class="mm-link">13.2 Partial Derivatives</a></li> <li><a href="/Classes/CalcIII/PartialDerivInterp.aspx" class="mm-link">13.3 Interpretations of Partial Derivatives</a></li> <li><a href="/Classes/CalcIII/HighOrderPartialDerivs.aspx" class="mm-link">13.4 Higher Order Partial Derivatives</a></li> <li><a href="/Classes/CalcIII/Differentials.aspx" class="mm-link">13.5 Differentials</a></li> <li><a href="/Classes/CalcIII/ChainRule.aspx" class="mm-link">13.6 Chain Rule</a></li> <li><a href="/Classes/CalcIII/DirectionalDeriv.aspx" class="mm-link">13.7 Directional Derivatives</a></li> </ul> </li> <li><a href="/Classes/CalcIII/PartialDerivAppsIntro.aspx" class="mm-link">14. Applications of Partial Derivatives</a> <ul> <li><a href="/Classes/CalcIII/TangentPlanes.aspx" class="mm-link">14.1 Tangent Planes and Linear Approximations</a></li> <li><a href="/Classes/CalcIII/GradientVectorTangentPlane.aspx" class="mm-link">14.2 Gradient Vector, Tangent Planes and Normal Lines</a></li> <li><a href="/Classes/CalcIII/RelativeExtrema.aspx" class="mm-link">14.3 Relative Minimums and Maximums</a></li> <li><a href="/Classes/CalcIII/AbsoluteExtrema.aspx" class="mm-link">14.4 Absolute Minimums and Maximums</a></li> <li><a href="/Classes/CalcIII/LagrangeMultipliers.aspx" class="mm-link">14.5 Lagrange Multipliers</a></li> </ul> </li> <li><a href="/Classes/CalcIII/MultipleIntegralsIntro.aspx" class="mm-link">15. Multiple Integrals</a> <ul> <li><a href="/Classes/CalcIII/DoubleIntegrals.aspx" class="mm-link">15.1 Double Integrals</a></li> <li><a href="/Classes/CalcIII/IteratedIntegrals.aspx" class="mm-link">15.2 Iterated Integrals</a></li> <li><a href="/Classes/CalcIII/DIGeneralRegion.aspx" class="mm-link">15.3 Double Integrals over General Regions</a></li> <li><a href="/Classes/CalcIII/DIPolarCoords.aspx" class="mm-link">15.4 Double Integrals in Polar Coordinates</a></li> <li><a href="/Classes/CalcIII/TripleIntegrals.aspx" class="mm-link">15.5 Triple Integrals</a></li> <li><a href="/Classes/CalcIII/TICylindricalCoords.aspx" class="mm-link">15.6 Triple Integrals in Cylindrical Coordinates</a></li> <li><a href="/Classes/CalcIII/TISphericalCoords.aspx" class="mm-link">15.7 Triple Integrals in Spherical Coordinates</a></li> <li><a href="/Classes/CalcIII/ChangeOfVariables.aspx" class="mm-link">15.8 Change of Variables</a></li> <li><a href="/Classes/CalcIII/SurfaceArea.aspx" class="mm-link">15.9 Surface Area</a></li> <li><a href="/Classes/CalcIII/Area_Volume.aspx" class="mm-link">15.10 Area and Volume Revisited</a></li> </ul> </li> <li><a href="/Classes/CalcIII/LineIntegralsIntro.aspx" class="mm-link">16. Line Integrals</a> <ul> <li><a href="/Classes/CalcIII/VectorFields.aspx" class="mm-link">16.1 Vector Fields</a></li> <li><a href="/Classes/CalcIII/LineIntegralsPtI.aspx" class="mm-link">16.2 Line Integrals - Part I</a></li> <li><a href="/Classes/CalcIII/LineIntegralsPtII.aspx" class="mm-link">16.3 Line Integrals - Part II</a></li> <li><a href="/Classes/CalcIII/LineIntegralsVectorFields.aspx" class="mm-link">16.4 Line Integrals of Vector Fields</a></li> <li><a href="/Classes/CalcIII/FundThmLineIntegrals.aspx" class="mm-link">16.5 Fundamental Theorem for Line Integrals</a></li> <li><a href="/Classes/CalcIII/ConservativeVectorField.aspx" class="mm-link">16.6 Conservative Vector Fields</a></li> <li><a href="/Classes/CalcIII/GreensTheorem.aspx" class="mm-link">16.7 Green's Theorem</a></li> </ul> </li> <li><a href="/Classes/CalcIII/SurfaceIntegralsIntro.aspx" class="mm-link">17.Surface Integrals</a> <ul> <li><a href="/Classes/CalcIII/CurlDivergence.aspx" class="mm-link">17.1 Curl and Divergence</a></li> <li><a href="/Classes/CalcIII/ParametricSurfaces.aspx" class="mm-link">17.2 Parametric Surfaces</a></li> <li><a href="/Classes/CalcIII/SurfaceIntegrals.aspx" class="mm-link">17.3 Surface Integrals</a></li> <li><a href="/Classes/CalcIII/SurfIntVectorField.aspx" class="mm-link">17.4 Surface Integrals of Vector Fields</a></li> <li><a href="/Classes/CalcIII/StokesTheorem.aspx" class="mm-link">17.5 Stokes' Theorem</a></li> <li><a href="/Classes/CalcIII/DivergenceTheorem.aspx" class="mm-link">17.6 Divergence Theorem</a></li> </ul> </li> </ul> </li> <li><a href="/Classes/DE/DE.aspx" class="mm-link">Differential Equations</a> <ul> <li><a href="/Classes/DE/IntroBasic.aspx" class="mm-link">1. Basic Concepts</a> <ul> <li><a href="/Classes/DE/Definitions.aspx" class="mm-link">1.1 Definitions</a></li> <li><a href="/Classes/DE/DirectionFields.aspx" class="mm-link">1.2 Direction Fields</a></li> <li><a href="/Classes/DE/FinalThoughts.aspx" class="mm-link">1.3 Final Thoughts</a></li> </ul> </li> <li><a href="/Classes/DE/IntroFirstOrder.aspx" class="mm-link">2. First Order DE's</a> <ul> <li><a href="/Classes/DE/Linear.aspx" class="mm-link">2.1 Linear Equations</a></li> <li><a href="/Classes/DE/Separable.aspx" class="mm-link">2.2 Separable Equations</a></li> <li><a href="/Classes/DE/Exact.aspx" class="mm-link">2.3 Exact Equations</a></li> <li><a href="/Classes/DE/Bernoulli.aspx" class="mm-link">2.4 Bernoulli Differential Equations</a></li> <li><a href="/Classes/DE/Substitutions.aspx" class="mm-link">2.5 Substitutions</a></li> <li><a href="/Classes/DE/IoV.aspx" class="mm-link">2.6 Intervals of Validity</a></li> <li><a href="/Classes/DE/Modeling.aspx" class="mm-link">2.7 Modeling with First Order DE's</a></li> <li><a href="/Classes/DE/EquilibriumSolutions.aspx" class="mm-link">2.8 Equilibrium Solutions</a></li> <li><a href="/Classes/DE/EulersMethod.aspx" class="mm-link">2.9 Euler's Method</a></li> </ul> </li> <li><a href="/Classes/DE/IntroSecondOrder.aspx" class="mm-link">3. Second Order DE's</a> <ul> <li><a href="/Classes/DE/SecondOrderConcepts.aspx" class="mm-link">3.1 Basic Concepts</a></li> <li><a href="/Classes/DE/RealRoots.aspx" class="mm-link">3.2 Real & Distinct Roots</a></li> <li><a href="/Classes/DE/ComplexRoots.aspx" class="mm-link">3.3 Complex Roots</a></li> <li><a href="/Classes/DE/RepeatedRoots.aspx" class="mm-link">3.4 Repeated Roots</a></li> <li><a href="/Classes/DE/ReductionofOrder.aspx" class="mm-link">3.5 Reduction of Order</a></li> <li><a href="/Classes/DE/FundamentalSetsofSolutions.aspx" class="mm-link">3.6 Fundamental Sets of Solutions</a></li> <li><a href="/Classes/DE/Wronskian.aspx" class="mm-link">3.7 More on the Wronskian</a></li> <li><a href="/Classes/DE/NonhomogeneousDE.aspx" class="mm-link">3.8 Nonhomogeneous Differential Equations</a></li> <li><a href="/Classes/DE/UndeterminedCoefficients.aspx" class="mm-link">3.9 Undetermined Coefficients</a></li> <li><a href="/Classes/DE/VariationofParameters.aspx" class="mm-link">3.10 Variation of Parameters</a></li> <li><a href="/Classes/DE/Vibrations.aspx" class="mm-link">3.11 Mechanical Vibrations</a></li> </ul> </li> <li><a href="/Classes/DE/LaplaceIntro.aspx" class="mm-link">4. Laplace Transforms</a> <ul> <li><a href="/Classes/DE/LaplaceDefinition.aspx" class="mm-link">4.1 The Definition</a></li> <li><a href="/Classes/DE/LaplaceTransforms.aspx" class="mm-link">4.2 Laplace Transforms</a></li> <li><a href="/Classes/DE/InverseTransforms.aspx" class="mm-link">4.3 Inverse Laplace Transforms</a></li> <li><a href="/Classes/DE/StepFunctions.aspx" class="mm-link">4.4 Step Functions</a></li> <li><a href="/Classes/DE/IVPWithLaplace.aspx" class="mm-link">4.5 Solving IVP's with Laplace Transforms</a></li> <li><a href="/Classes/DE/IVPWithNonConstantCoefficient.aspx" class="mm-link">4.6 Nonconstant Coefficient IVP's</a></li> <li><a href="/Classes/DE/IVPWithStepFunction.aspx" class="mm-link">4.7 IVP's With Step Functions</a></li> <li><a href="/Classes/DE/DiracDeltaFunction.aspx" class="mm-link">4.8 Dirac Delta Function</a></li> <li><a href="/Classes/DE/ConvolutionIntegrals.aspx" class="mm-link">4.9 Convolution Integrals</a></li> <li><a href="/Classes/DE/Laplace_Table.aspx" class="mm-link">4.10 Table Of Laplace Transforms</a></li> </ul> </li> <li><a href="/Classes/DE/SystemsIntro.aspx" class="mm-link">5. Systems of DE's</a> <ul> <li><a href="/Classes/DE/LA_Systems.aspx" class="mm-link">5.1 Review : Systems of Equations</a></li> <li><a href="/Classes/DE/LA_Matrix.aspx" class="mm-link">5.2 Review : Matrices & Vectors</a></li> <li><a href="/Classes/DE/LA_Eigen.aspx" class="mm-link">5.3 Review : Eigenvalues & Eigenvectors</a></li> <li><a href="/Classes/DE/SystemsDE.aspx" class="mm-link">5.4 Systems of Differential Equations</a></li> <li><a href="/Classes/DE/SolutionsToSystems.aspx" class="mm-link">5.5 Solutions to Systems</a></li> <li><a href="/Classes/DE/PhasePlane.aspx" class="mm-link">5.6 Phase Plane</a></li> <li><a href="/Classes/DE/RealEigenvalues.aspx" class="mm-link">5.7 Real Eigenvalues</a></li> <li><a href="/Classes/DE/ComplexEigenvalues.aspx" class="mm-link">5.8 Complex Eigenvalues</a></li> <li><a href="/Classes/DE/RepeatedEigenvalues.aspx" class="mm-link">5.9 Repeated Eigenvalues</a></li> <li><a href="/Classes/DE/NonhomogeneousSystems.aspx" class="mm-link">5.10 Nonhomogeneous Systems</a></li> <li><a href="/Classes/DE/SystemsLaplace.aspx" class="mm-link">5.11 Laplace Transforms</a></li> <li><a href="/Classes/DE/SystemsModeling.aspx" class="mm-link">5.12 Modeling</a></li> </ul> </li> <li><a href="/Classes/DE/SeriesIntro.aspx" class="mm-link">6. Series Solutions to DE's</a> <ul> <li><a href="/Classes/DE/PowerSeries.aspx" class="mm-link">6.1 Review : Power Series</a></li> <li><a href="/Classes/DE/TaylorSeries.aspx" class="mm-link">6.2 Review : Taylor Series</a></li> <li><a href="/Classes/DE/SeriesSolutions.aspx" class="mm-link">6.3 Series Solutions</a></li> <li><a href="/Classes/DE/EulerEquations.aspx" class="mm-link">6.4 Euler Equations</a></li> </ul> </li> <li><a href="/Classes/DE/IntroHigherOrder.aspx" class="mm-link">7. Higher Order Differential Equations</a> <ul> <li><a href="/Classes/DE/HOBasicConcepts.aspx" class="mm-link">7.1 Basic Concepts for <em>n</em><sup>th</sup> Order Linear Equations</a></li> <li><a href="/Classes/DE/HOHomogeneousDE.aspx" class="mm-link">7.2 Linear Homogeneous Differential Equations</a></li> <li><a href="/Classes/DE/HOUndeterminedCoeff.aspx" class="mm-link">7.3 Undetermined Coefficients</a></li> <li><a href="/Classes/DE/HOVariationOfParam.aspx" class="mm-link">7.4 Variation of Parameters</a></li> <li><a href="/Classes/DE/HOLaplaceTransforms.aspx" class="mm-link">7.5 Laplace Transforms</a></li> <li><a href="/Classes/DE/HOSystems.aspx" class="mm-link">7.6 Systems of Differential Equations</a></li> <li><a href="/Classes/DE/HOSeries.aspx" class="mm-link">7.7 Series Solutions</a></li> </ul> </li> <li><a href="/Classes/DE/IntroBVP.aspx" class="mm-link">8. Boundary Value Problems & Fourier Series</a> <ul> <li><a href="/Classes/DE/BoundaryValueProblem.aspx" class="mm-link">8.1 Boundary Value Problems</a></li> <li><a href="/Classes/DE/BVPEvals.aspx" class="mm-link">8.2 Eigenvalues and Eigenfunctions</a></li> <li><a href="/Classes/DE/PeriodicOrthogonal.aspx" class="mm-link">8.3 Periodic Functions & Orthogonal Functions</a></li> <li><a href="/Classes/DE/FourierSineSeries.aspx" class="mm-link">8.4 Fourier Sine Series</a></li> <li><a href="/Classes/DE/FourierCosineSeries.aspx" class="mm-link">8.5 Fourier Cosine Series</a></li> <li><a href="/Classes/DE/FourierSeries.aspx" class="mm-link">8.6 Fourier Series</a></li> <li><a href="/Classes/DE/ConvergenceFourierSeries.aspx" class="mm-link">8.7 Convergence of Fourier Series</a></li> </ul> </li> <li><a href="/Classes/DE/IntroPDE.aspx" class="mm-link">9. Partial Differential Equations </a> <ul> <li><a href="/Classes/DE/TheHeatEquation.aspx" class="mm-link">9.1 The Heat Equation</a></li> <li><a href="/Classes/DE/TheWaveEquation.aspx" class="mm-link">9.2 The Wave Equation</a></li> <li><a href="/Classes/DE/PDETerminology.aspx" class="mm-link">9.3 Terminology</a></li> <li><a href="/Classes/DE/SeparationofVariables.aspx" class="mm-link">9.4 Separation of Variables</a></li> <li><a href="/Classes/DE/SolvingHeatEquation.aspx" class="mm-link">9.5 Solving the Heat Equation</a></li> <li><a href="/Classes/DE/HeatEqnNonZero.aspx" class="mm-link">9.6 Heat Equation with Non-Zero Temperature Boundaries</a></li> <li><a href="/Classes/DE/LaplacesEqn.aspx" class="mm-link">9.7 Laplace's Equation</a></li> <li><a href="/Classes/DE/VibratingString.aspx" class="mm-link">9.8 Vibrating String</a></li> <li><a href="/Classes/DE/PDESummary.aspx" class="mm-link">9.9 Summary of Separation of Variables</a></li> </ul> </li> </ul> </li> <li><span>Extras</span></li> <li><a href="/Extras/AlgebraTrigReview/AlgebraTrig.aspx" class="mm-link">Algebra & Trig Review</a> <ul> <li><a href="/Extras/AlgebraTrigReview/AlgebraIntro.aspx" class="mm-link">1. Algebra</a> <ul> <li><a href="/Extras/AlgebraTrigReview/Exponents.aspx" class="mm-link">1.1 Exponents </a></li> <li><a href="/Extras/AlgebraTrigReview/AbsoluteValue.aspx" class="mm-link">1.2 Absolute Value</a></li> <li><a href="/Extras/AlgebraTrigReview/Radicals.aspx" class="mm-link">1.3 Radicals</a></li> <li><a href="/Extras/AlgebraTrigReview/Rationalizing.aspx" class="mm-link">1.4 Rationalizing </a></li> <li><a href="/Extras/AlgebraTrigReview/Functions.aspx" class="mm-link">1.5 Functions </a></li> <li><a href="/Extras/AlgebraTrigReview/MultPoly.aspx" class="mm-link">1.6 Multiplying Polynomials</a></li> <li><a href="/Extras/AlgebraTrigReview/Factoring.aspx" class="mm-link">1.7 Factoring</a></li> <li><a href="/Extras/AlgebraTrigReview/SimpRatExp.aspx" class="mm-link">1.8 Simplifying Rational Expressions</a></li> <li><a href="/Extras/AlgebraTrigReview/Graphing.aspx" class="mm-link">1.9 Graphing and Common Graphs</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveEqnPtI.aspx" class="mm-link">1.10 Solving Equations, Part I</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveEqnPtII.aspx" class="mm-link">1.11 Solving Equations, Part II</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveSystems.aspx" class="mm-link">1.12 Solving Systems of Equations</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveIneq.aspx" class="mm-link">1.13 Solving Inequalities</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveAbsValue.aspx" class="mm-link">1.14 Absolute Value Equations and Inequalities</a></li> </ul> </li> <li><a href="/Extras/AlgebraTrigReview/TrigIntro.aspx" class="mm-link">2. Trigonometry</a> <ul> <li><a href="/Extras/AlgebraTrigReview/TrigFunctions.aspx" class="mm-link">2.1 Trig Function Evaluation</a></li> <li><a href="/Extras/AlgebraTrigReview/TrigGraphs.aspx" class="mm-link">2.2 Graphs of Trig Functions</a></li> <li><a href="/Extras/AlgebraTrigReview/TrigFormulas.aspx" class="mm-link">2.3 Trig Formulas</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveTrigEqn.aspx" class="mm-link">2.4 Solving Trig Equations</a></li> <li><a href="/Extras/AlgebraTrigReview/InverseTrig.aspx" class="mm-link">2.5 Inverse Trig Functions</a></li> </ul> </li> <li><a href="/Extras/AlgebraTrigReview/ExpLogIntro.aspx" class="mm-link">3. Exponentials & Logarithms</a> <ul> <li><a href="/Extras/AlgebraTrigReview/ExponentialFcns.aspx" class="mm-link">3.1 Basic Exponential Functions</a></li> <li><a href="/Extras/AlgebraTrigReview/LogarithmFcns.aspx" class="mm-link">3.2 Basic Logarithm Functions</a></li> <li><a href="/Extras/AlgebraTrigReview/LogProperties.aspx" class="mm-link">3.3 Logarithm Properties</a></li> <li><a href="/Extras/AlgebraTrigReview/SimpLogs.aspx" class="mm-link">3.4 Simplifying Logarithms</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveExpEqn.aspx" class="mm-link">3.5 Solving Exponential Equations</a></li> <li><a href="/Extras/AlgebraTrigReview/SolveLogEqn.aspx" class="mm-link">3.6 Solving Logarithm Equations</a></li> </ul> </li> </ul> </li> <li><a href="/Extras/CommonErrors/CommonMathErrors.aspx" class="mm-link">Common Math Errors</a> <ul> <li><a href="/Extras/CommonErrors/GeneralErrors.aspx" class="mm-link">1. General Errors</a> </li> <li><a href="/Extras/CommonErrors/AlgebraErrors.aspx" class="mm-link">2. Algebra Errors</a> </li> <li><a href="/Extras/CommonErrors/TrigErrors.aspx" class="mm-link">3. Trig Errors</a> </li> <li><a href="/Extras/CommonErrors/CommonErrors.aspx" class="mm-link">4. Common Errors</a> </li> <li><a href="/Extras/CommonErrors/CalculusErrors.aspx" class="mm-link">5. Calculus Errors</a> </li> </ul> </li> <li><a href="/Extras/ComplexPrimer/ComplexNumbers.aspx" class="mm-link">Complex Number Primer</a> <ul> <li><a href="/Extras/ComplexPrimer/Definition.aspx" class="mm-link">1. The Definition</a> </li> <li><a href="/Extras/ComplexPrimer/Arithmetic.aspx" class="mm-link">2. Arithmetic</a> </li> <li><a href="/Extras/ComplexPrimer/ConjugateModulus.aspx" class="mm-link">3. Conjugate and Modulus</a> </li> <li><a href="/Extras/ComplexPrimer/Forms.aspx" class="mm-link">4. Polar and Exponential Forms</a> </li> <li><a href="/Extras/ComplexPrimer/Roots.aspx" class="mm-link">5. Powers and Roots</a> </li> </ul> </li> <li><a href="/Extras/StudyMath/HowToStudyMath.aspx" class="mm-link">How To Study Math</a> <ul> <li><a href="/Extras/StudyMath/GeneralTips.aspx" class="mm-link">1. General Tips</a> </li> <li><a href="/Extras/StudyMath/TakingNotes.aspx" class="mm-link">2. Taking Notes</a> </li> <li><a href="/Extras/StudyMath/GettingHelp.aspx" class="mm-link">3. Getting Help</a> </li> <li><a href="/Extras/StudyMath/Homework.aspx" class="mm-link">4. Doing Homework</a> </li> <li><a href="/Extras/StudyMath/ProblemSolving.aspx" class="mm-link">5. Problem Solving</a> </li> <li><a href="/Extras/StudyMath/StudyForExam.aspx" class="mm-link">6. Studying For an Exam</a> </li> <li><a href="/Extras/StudyMath/TakingExam.aspx" class="mm-link">7. Taking an Exam</a> </li> <li><a href="/Extras/StudyMath/Errors.aspx" class="mm-link">8. Learn From Your Errors</a> </li> </ul> </li> <li><span>Misc Links</span></li> <li><a href="/contact.aspx" class="mm-link">Contact Me</a></li> <li><a href="/links.aspx" class="mm-link">Links</a></li> <li><a href="/mathjax.aspx" class="mm-link">MathJax Help and Configuration</a></li> <li><a href="/privacy.aspx" class="mm-link">Privacy Statement</a></li> <li><a href="/help.aspx" class="mm-link">Site Help & FAQ</a></li> <li><a href="/terms.aspx" class="mm-link">Terms of Use</a></li> </ul> </nav> <div class="top-info-bar"> <span id="mobile-title" class="mobile-header-title header-title">Paul's Online Notes</span> <br /> <span class="top-menu-breadcrumb"> <a href="/" id="breadcrumb_home_link" title="Go to Main Page">Home</a> <span id="breadcrumb_h_b_sep_span">/ </span> <a href="/Classes/CalcI/CalcI.aspx" id="breadcrumb_book_link" title="Go to Book Introduction">Calculus I</a> <span id="breadcrumb_b_c_sep_span">/ </span> <a href="/Classes/CalcI/DerivativeIntro.aspx" id="breadcrumb_chapter_link" title="Go to Chapter Introduction">Derivatives</a> <span id="breadcrumb_section_span"> / Product and Quotient Rule</span> </span> </div> <div id="nav_links" class="top-nav-bar"> <a href="/Classes/CalcI/DiffFormulas.aspx" id="nav_links_prev_section" title="Goto Previous Section : Differentiation Formulas"><span class="top-menu-prev fas fa-chevron-left"></span><span class="nav_desktop_extra_pn"> Prev. Section</span></a> <div class="top-nav-bar-link-spacer"></div> <span id="nav_current_notes">Notes</span> <a href="/Problems/CalcI/ProductQuotientRule.aspx" id="nav_links_practice" title="Go to Practice Problems for current topic.">Practice<span class="nav_desktop_extra"> Problems</span></a> <a href="/ProblemsNS/CalcI/ProductQuotientRule.aspx" id="nav_links_asgn" title="Go to Assignment Problems for current topic.">Assignment<span class="nav_desktop_extra"> Problems</span></a> <div class="top-nav-bar-link-spacer"></div> <a href="/Classes/CalcI/DiffTrigFcns.aspx" id="nav_links_next_section" title="Goto Next Section : Derivatives of Trig Functions"><span class="nav_desktop_extra_pn"> Next Section </span><span class="top-menu-next fas fa-chevron-right"></span></a> </div> <div class="header-divider"></div> <div class="content"> <span id="SHLink_NoteMobile" class="SH-Link content-note-link-mobile">Show Mobile Notice</span> <span id="SHImg_NoteMobile" class="fas fa-caret-right SH-padding content-note-link-mobile" aria-hidden="true"></span> <span id="SHALink_S_Note" class="SH-Link SH-Hide SH-Bracket">Show All Notes</span> <span id="SHALink_H_Note" class="SH-Link SH-Hide SH-Bracket">Hide All Notes</span> <div id="SHObj_NoteMobile" class="content-note-container content-note-container-mobile"> <div class="content-note-header">Mobile Notice</div> <div class="content-note">You appear to be on a device with a "narrow" screen width (<em>i.e.</em> you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.</div> </div> <form method="post" action="./ProductQuotientRule.aspx" id="ctl00"> <div class="aspNetHidden"> <input type="hidden" name="__VIEWSTATE" id="__VIEWSTATE" value="31j0ajSujmU1EM/Sc38DwdT2Why5+hbmSDsR/zV25lPwHSEYbtPSUBIwaST9N/x+cvux518JdSJIwUcEDLh6b+2JGonfPtlOYv7PN/L5IEOgCmL97ccnc7u82JXFpRHeUVGJP9xlPglEnc6AdvzI7bT35PPJR86gJsOyBx36L6d4e/XyArm46U01GHLbuwR3zh5roCL7JPR/4SP6fPE8EIBT4UQWpi7llPAhcQjRixWdKjKxrD1FJQo6kKQrQQ+wumPpEyw8cfAKXFDG1ZuG1u02Yz99zRJYryKFAlDwsyHVeJsqwxiK/0eWL7nIRB7Xcdwqob/XxtmUaA/NXW8sOuD0Q3Hrh/qqD39bF787s0qcptSNfCCw0jpRN0Jp4HVWZRHteNlJZbDQw2ABgkOTfAn5Po83VcrgHYbSyA73U95TtOPrqoXqhuxix00QQNu4iBQ2eAzN+vmPRyHZFnmqyzj9y0bWucK8lUsnvUbGCsZHWR6YR8x6O/MW5xBCP0uqxYo6yWE75hkqxFHNVfXYtz+Ed5yHrVbbSXuUS2jWAUFFYkXLBCzqWcpxuNV6GPz6Y589IEFM+1wWAnfit5x9IeHk8Ljawv5WduooGD4zohglB0bjI2gHAy7unWc3EpteSXXZR3IGqXT107msh/7NB1Jk9NAHws8hkq/cl558V8uydH+p1eGR+ci9j9NF5TOOQl/FhhUFY3HRbbYtXJK5dKGCCcIAik7EkOtGThiKW4pDCDIqp3oHxbMN/qD73H85WJZCFuEjFxC62AJG1+jUpOgEdz4YbKAtaPYaNl+3RO6orw7HOTuIoLZeWtSdLJZ+emGskBgWd8x9k8ACGItZSrgv2fN3PxkrAYrLtrHgrbltjunDBJ7Rzv+4MYmWnZPvszcb9njNbdDQpOl65rY5exCo54Nbg3mgHqEyHlyN3nA3m/a4qYSGaNbna+M8EAwkTDd4zJRy0amaIfYFZlOmqkfzuUjHT0eHESMW2oeu/J8rNpTjC9QmVDtZjAp00Y5+008/4twXMfQk2ljmUT4i+H3EmTn5JPlr2PKWP6p1G0YipToE0GWt3KFvtS8H5hCeTT0A1EFAsEyiedheXmrExPNr7/68L6vkQumHqrc+K0goWr0+1/CPwYIQnP+8UdqXDnwIHQ+ec1pJptmROv6of/WUwdqTZbt4mE33SMSw7dq8fSIaImxJWKMdUujWezH43pYp8N9BXzw4CyuYll02UMGv2UbrXI8SqMvjPgFeUIhv/iLpfAFA/I3Jx6581SjxcvUIBQnqKYwkR54Lofl5vKFks3cCx9XkciaMxwVw7x056D+ffNYvqfNRI9Alkgf5KWpGgCdPAXTViB+8xYchNPsd7rbqIoBhiehUPmp9THNJ12SgrCW95vp47/6hZkzwaK+j4euTBafzOexRuT/jj6y1nllYLWHV1kSTVASbpyxeKayxgeheNC8do7bJ+nlU8TaWfi4GM4V2unX9WzELTo8Hqa/HN7JKogG7ig5XttsOqR5d/C+ZRVIjOmbmAeZcH7eHPuo8hdsfLoVLbjzkNAD0/kbJl4mWWwhLSnhP51PAzSfXu7lnmg3rGhMzrnq3N5L4FKIqGClK0lzxs1M7ELXL4rM36+SXY8dmWDtvXhlaX2Ol0I2TRGQTPqLh7HC/KhgNlKRB8eMXwOM7jJjYH1VlTrnoWW8s1mMssc2bJr5gHMm3vRghV7ubBmYbngL4rp2a5HD5G+uP7Ut7BowU/d6Rw8sj6OKyjxa5fsFgKdWFVygw0gUpebfmZM7MyW3OFfUONXQ/39fTqN0NVXGST/G+s63bPPe1yTiqO/e7d6oIDlExS3Ef4n1Cm72xk9kVOBud0/OKkHIAmkNl4lTCg9bBgi66dYNTjsggmcyfRPsa+rXm+p/iCXKChI8cVu5Lrg/j5n3ORPatpFi61xJKb5Cm+jlo7WKBlOGSZH/18dqYJOWdK/M2E9U6ATl3wOwKuwsGAtatDj1mJNYbVutDn03E3707oy/D2cd7mt/VGf2odJYrhStPuVuGoULLOXWeXocRnlMjsa1w0ML/dxb+6wP3OJQ42gzY1X+77i0glQXLb1VrSokJ+mktWIgP0i9M+Hh3MIkzMiOfFRyMfRWt4xpFHR3A/MFJVrgJweETdl98lOtt7XKmeRc16P00IrE2TTYdbu1c+Ns00Ot44ocm4Pb887uAQJKOcb32iTCo7bQ84jl0uEDdrHBwhnh4DcD1bzqgs37htr8Jg8IEi2w8J0M2bnvQsJGE7lrbs93zCob2IsrZbFkQvImnW2/RMngYGQ1NsIyw0byIKvrRI4i4T+GpfdE3a6qrmcFWfXa2Wlm+jNMHq1C+aOFziFyOaN/GgKPKvtaEu9z0GR94TjtKcDB3F8vyuaVR1s0NyAyp1kdBefhGtlQ5i0bMnC1JIHk3Au65bZKGnqaH+C7uuORhyySM6AVuh3sEEi5uQvns17H6EqXlJG4PNCNmbp24d37qBeZ6NPiK/1bU7OVkLTbp+U67aZAwgYrQqo3dxMWlpOaSw/IXVgel5VfcT2n7tgZ8AMZ6XTQzounrybmoFL4+4vqG0sFQBURwlNKAAb3QwZ0KUxVCcCTsSsUaxgHWQtZk/Yh7c/0aQfHSappcTxifZ0LgrzOVEgKPWT4CJMRFmHRYALzCN+zI/dbehcZbGmEI58Hw7WbukIib4ykg3ZRZHfgj6GTRBXbntpXAoSvMEmfePDwZEVEUBMihUijlg1GetodkssCAcXXQRIqENaCDrNpnI2ZZkyLt2tRPnxYXPtkqnRZ2uc9sivNpHKOd14igxESIEMThOiE0D9ihZHgman4iDI46SRQWnwfSsxsd+yp1X5YPYFJOnkDhGs6gq57BT4gAZ3Uh3sfJmpKajxnOpUao5T2Saq4u4tqFf2NbYzQQQXL4yf9bXgBwxYIfGe1gT/VF2bQbaMpwrianHdh37Zz8PTVhPiZgP0PwVh/lIbEaCM768URKi2G+JicUHuoi0bYLkJiLN6Q0eduw/WY58RPqWzRmPrXg0wfaN6ZHUnqeg0jkVlI30LC0EF96KRQUn5MKQGfu0GsVPABG8EbLYYWTQKDfxKh3Q+ycD6uACxl3aEIVbf7ZDLvnLPq76/ebFrudqey8/GxA/uSKFjVuCErQ0ZFvDq8drhZpAt7d/nJq9jRWKSQE/wQGltJBl0SEpUE+zq1WCm28m3CzXCSOKhu7Ub4sHqnaF4tUPzFXYgK95F46TSClusMvHcAWnoG/T+SUoq91UWC7oJBWG+SAVQFgSkBatFODhF77ayfKB+2lPAt1eTRg/l9Kj+2/QiG9LZtAhz2yeZMnjfF8mCAjOx78ZeP6bNM4TVsllTT2C+KC2uJdPCGpObNlXQkqmXIq4QLMiNxvOvVJ7dtJZL/egz5tu+ipD2c/ClyCeiXWPcTSdjXSgKhAh3tu4SLSeGpnyjFh1IsZ0Cbt51jmcrT8nJ4ZyQ8KrOQ/19EeOPVGVb80bf8G5HzNs9XFdHoiuAz4EDeixGp8tnYfItTZsfS3JremvIN33XoEyFRkne+xXX9ETaczqy8o9MZPGnRVLgMQFT+NJ5BMdi4pDeFck7rXxoXG6jXwBa5yOE65lVGMDcPTXFIbXeJ5PGMU/G67v0vO" /> </div> <div class="aspNetHidden"> <input type="hidden" name="__VIEWSTATEGENERATOR" id="__VIEWSTATEGENERATOR" value="0F5F4EDF" /> </div> </form> <h3>Section 3.4 : Product and Quotient Rule</h3> <p>In the previous section we noted that we had to be careful when differentiating products or quotients. It’s now time to look at products and quotients and see why.</p> <p>First let’s take a look at why we have to be careful with products and quotients. Suppose that we have the two functions $f\left( x \right) = {x^3}$ and $g\left( x \right) = {x^6}$. Let’s start by computing the derivative of the product of these two functions. This is easy enough to do directly.</p> \[{\left( {f\,g} \right)^\prime } = {\left( {{x^3}{x^6}} \right)^\prime } = {\left( {{x^9}} \right)^\prime } = 9{x^8}\] <p>Remember that on occasion we will drop the $\left( x \right)$ part on the functions to simplify notation somewhat. We’ve done that in the work above.</p> <p>Now, let’s try the following.</p> \[f'\left( x \right)g'\left( x \right) = \left( {3{x^2}} \right)\left( {6{x^5}} \right) = 18{x^7}\] <p>So, we can very quickly see that.</p> \[{\left( {f\,g} \right)^\prime } \ne f'\,g'\] <p>In other words, the derivative of a product is not the product of the derivatives.</p> <p>Using the same functions we can do the same thing for quotients.</p> \[{\left( {\frac{f}{g}} \right)^\prime } = {\left( {\frac{{{x^3}}}{{{x^6}}}} \right)^\prime } = {\left( {\frac{1}{{{x^3}}}} \right)^\prime } = {\left( {{x^{ - 3}}} \right)^\prime } = - 3{x^{ - 4}} = - \frac{3}{{{x^4}}}\] \[\frac{{f'\left( x \right)}}{{g'\left( x \right)}} = \frac{{3{x^2}}}{{6{x^5}}} = \frac{1}{{2{x^3}}}\] <p>So, again we can see that,</p> \[{\left( {\frac{f}{g}} \right)^\prime } \ne \frac{{f'}}{{g'}}\] <p>To differentiate products and quotients we have the <strong>Product Rule</strong> and the <strong>Quotient Rule</strong>.</p> <h4>Product Rule</h4> <div class="fact"> <p>If the two functions $f\left( x \right)$ and $g\left( x \right)$ are differentiable (<em>i.e.</em> the derivative exist) then the product is differentiable and,</p> \[{\left( {f\,g} \right)^\prime } = f'\,g + f\,g'\] </div> <p>The proof of the Product Rule is shown in the <a href="DerivativeProofs.aspx">Proof of Various Derivative Formulas</a> section of the Extras chapter.</p> <h4>Quotient Rule</h4> <div class="fact"> <p>If the two functions $f\left( x \right)$ and $g\left( x \right)$ are differentiable (<em>i.e.</em> the derivative exist) then the quotient is differentiable and,</p> \[{\left( {\frac{f}{g}} \right)^\prime } = \frac{{f'\,g - f\,g'}}{{{g^2}}}\] </div> <p>Note that the numerator of the quotient rule is very similar to the product rule so be careful to not mix the two up!</p> <p>The proof of the Quotient Rule is shown in the <a href="DerivativeProofs.aspx">Proof of Various Derivative Formulas</a> section of the Extras chapter.</p> <p>Let’s do a couple of examples of the product rule.</p> <a class="anchor" name="Deriv_PQRule_Ex1"></a> <div class="example"> <span class="example-title">Example 1</span> Differentiate each of the following functions. <ol class="example_parts_list"> <li>$y = \sqrt[3]{{{x^2}}}\left( {2x - {x^2}} \right)$</li> <li>$f\left( x \right) = \left( {6{x^3} - x} \right)\left( {10 - 20x} \right)$</li> </ol> <span id="SHALink_S_Soln1" class="SH-Link SH-All">Show All Solutions</span> <span id="SHALink_H_Soln1" class="SH-Link SH-All">Hide All Solutions</span> <div class="example-content"> <p>At this point there really aren’t a lot of reasons to use the product rule. As we noted in the previous section all we would need to do for either of these is to just multiply out the product and then differentiate.</p> <p>With that said we will use the product rule on these so we can see an example or two. As we add more functions to our repertoire and as the functions become more complicated the product rule will become more useful and in many cases required.</p> <span class="soln-list-item soln-list-subitem">a</span> $y = \sqrt[3]{{{x^2}}}\left( {2x - {x^2}} \right)$ <span id="SHLink_Soln1a" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln1a" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln1a" class="soln-content"> <p>Note that we took the derivative of this function in the previous <a href="DiffFormulas.aspx#Deriv_Form_Ex2a">section</a> and didn’t use the product rule at that point. We should however get the same result here as we did then.</p> <p>Now let’s do the problem here. There’s not really a lot to do here other than use the product rule. However, before doing that we should convert the radical to a fractional exponent as always.</p> \[y = {x^{\frac{2}{3}}}\left( {2x - {x^2}} \right)\] <p>Now let’s take the derivative. So, we take the derivative of the first function times the second then add on to that the first function times the derivative of the second function.</p> \[y' = \frac{2}{3}{x^{ - \frac{1}{3}}}\left( {2x - {x^2}} \right) + {x^{\frac{2}{3}}}\left( {2 - 2x} \right)\] <p>This is NOT what we got in the previous section for this derivative. However, with some simplification we can arrive at the same answer.</p> \[y' = \frac{4}{3}{x^{\frac{2}{3}}} - \frac{2}{3}{x^{\frac{5}{3}}} + 2{x^{\frac{2}{3}}} - 2{x^{\frac{5}{3}}} = \frac{{10}}{3}{x^{\frac{2}{3}}} - \frac{8}{3}{x^{\frac{5}{3}}}\] <p>This is what we got for an answer in the previous section so that is a good check of the product rule.</p> </div> <br /> <span class="soln-list-item soln-list-subitem">b</span> $f\left( x \right) = \left( {6{x^3} - x} \right)\left( {10 - 20x} \right)$ <span id="SHLink_Soln1b" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln1b" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln1b" class="soln-content"> <p>This one is actually easier than the previous one. Let’s just run it through the product rule.</p> \[\begin{align*}f'\left( x \right) & = \left( {18{x^2} - 1} \right)\left( {10 - 20x} \right) + \left( {6{x^3} - x} \right)\left( { - 20} \right)\\ & = - 480{x^3} + 180{x^2} + 40x - 10\end{align*}\] <p>Since it was easy to do we went ahead and simplified the results a little.</p> </div> </div> </div> <p>Let’s now work an example or two with the quotient rule. In this case, unlike the product rule examples, a couple of these functions will require the quotient rule in order to get the derivative. For the last two however, we can avoid the quotient rule if we’d like to as we’ll see.</p> <a class="anchor" name="Deriv_PQRule_Ex2"></a> <div class="example"> <span class="example-title">Example 2</span> Differentiate each of the following functions. <ol class="example_parts_list"> <li>$\displaystyle W\left( z \right) = \frac{{3z + 9}}{{2 - z}}$</li> <li>$\displaystyle h\left( x \right) = \frac{{4\sqrt x }}{{{x^2} - 2}}$</li> <li>$\displaystyle f\left( x \right) = \frac{4}{{{x^6}}}$</li> <li>$\displaystyle y = \frac{{{w^6}}}{5}$</li> </ol> <span id="SHALink_S_Soln2" class="SH-Link SH-All">Show All Solutions</span> <span id="SHALink_H_Soln2" class="SH-Link SH-All">Hide All Solutions</span> <div class="example-content"> <span class="soln-list-item soln-list-subitem soln-list-subitem-padding-20">a</span> $\displaystyle W\left( z \right) = \frac{{3z + 9}}{{2 - z}}$ <span id="SHLink_Soln2a" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln2a" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln2a" class="soln-content"> <p>There isn’t a lot to do here other than to use the quotient rule. Here is the work for this function.</p> \[\begin{align*}W'\left( z \right) & = \frac{{3\left( {2 - z} \right) - \left( {3z + 9} \right)\left( { - 1} \right)}}{{{{\left( {2 - z} \right)}^2}}}\\ & = \frac{{15}}{{{{\left( {2 - z} \right)}^2}}}\end{align*}\] </div> <br /> <span class="soln-list-item soln-list-subitem soln-list-subitem-padding-20">b</span> $\displaystyle h\left( x \right) = \frac{{4\sqrt x }}{{{x^2} - 2}}$ <span id="SHLink_Soln2b" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln2b" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln2b" class="soln-content"> <p>Again, not much to do here other than use the quotient rule. Don’t forget to convert the square root into a fractional exponent.</p> \[\begin{align*}h'\left( x \right) & = \frac{{4\left( {{\textstyle{1 \over 2}}} \right){x^{ - \frac{1}{2}}}\left( {{x^2} - 2} \right) - 4{x^{\frac{1}{2}}}\left( {2x} \right)}}{{{{\left( {{x^2} - 2} \right)}^2}}}\\ & = \frac{{2{x^{\frac{3}{2}}} - 4{x^{ - \frac{1}{2}}} - 8{x^{\frac{3}{2}}}}}{{{{\left( {{x^2} - 2} \right)}^2}}}\\ & = \frac{{ - 6{x^{\frac{3}{2}}} - 4{x^{ - \frac{1}{2}}}}}{{{{\left( {{x^2} - 2} \right)}^2}}}\end{align*}\] </div> <br /> <span class="soln-list-item soln-list-subitem soln-list-subitem-padding-20">c</span> $\displaystyle f\left( x \right) = \frac{4}{{{x^6}}}$ <span id="SHLink_Soln2c" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln2c" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln2c" class="soln-content"> <p>It seems strange to have this one here rather than being the first part of this example given that it definitely appears to be easier than any of the previous two. In fact, it is easier. There is a point to doing it here rather than first. In this case there are two ways to do compute this derivative. There is an easy way and a hard way and in this case the hard way is the quotient rule. That’s the point of this example.</p> <p>Let’s do the quotient rule and see what we get.</p> \[f'\left( x \right) = \frac{{\left( 0 \right)\left( {{x^6}} \right) - 4\left( {6{x^5}} \right)}}{{{{\left( {{x^6}} \right)}^2}}} = \frac{{ - 24{x^5}}}{{{x^{12}}}} = - \frac{{24}}{{{x^7}}}\] <p>Now, that was the “hard” way. So, what was so hard about it? Well actually it wasn’t that hard, there is just an easier way to do it that’s all. However, having said that, a common mistake here is to do the derivative of the numerator (a constant) incorrectly. For some reason many people will give the derivative of the numerator in these kinds of problems as a 1 instead of 0! Also, there is some simplification that needs to be done in these kinds of problems if you do the quotient rule.</p> <p>The easy way is to do what we did in the previous section.</p> \[f\left( x \right) = 4{x^{ - 6}} \hspace{0.5in} f'\left( x \right) = - 24{x^{ - 7}} = - \frac{{24}}{{{x^7}}}\] <p>Either way will work, but I’d rather take the easier route if I had the choice.</p> </div> <br /> <span class="soln-list-item soln-list-subitem">d</span> $\displaystyle y = \frac{{{w^6}}}{5}$ <span id="SHLink_Soln2d" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln2d" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln2d" class="soln-content"> <p>This problem also seems a little out of place. However, it is here again to make a point. Do not confuse this with a quotient rule problem. While you can do the quotient rule on this function there is no reason to use the quotient rule on this. Simply rewrite the function as</p> \[y = \frac{1}{5}{w^6}\] <p>and differentiate as always.</p> \[y' = \frac{6}{5}{w^5}\] </div> </div> </div> <p>Finally, let’s not forget about our applications of derivatives.</p> <a class="anchor" name="Deriv_PQRule_Ex3"></a> <div class="example"> <span class="example-title">Example 3</span> Suppose that the amount of air in a balloon at any time $t$ is given by \[V\left( t \right) = \frac{{6\sqrt[3]{t}}}{{4t + 1}}\] <p>Determine if the balloon is being filled with air or being drained of air at $t = 8$.</p> <div class="example-content"> <span id="SHLink_Soln3" class="SH-Link soln-title">Show Solution</span> <span id="SHImg_Soln3" class="fas fa-caret-right" aria-hidden="true"></span> <div id="SHObj_Soln3" class="soln-content"> <p>If the balloon is being filled with air then the volume is increasing and if it’s being drained of air then the volume will be decreasing. In other words, we need to get the derivative so that we can determine the rate of change of the volume at $t = 8$<em>.</em></p> <p>This will require the quotient rule.</p> \[\begin{align*} {V}'\left( t \right) & =\frac{2{{t}^{-\frac{2}{3}}}\left( 4t+1 \right)-6{{t}^{\frac{1}{3}}}\left( 4 \right)}{{{\left( 4t+1 \right)}^{2}}} \\ & =\frac{-16{{t}^{\frac{1}{3}}}+2{{t}^{-\frac{2}{3}}}}{{{\left( 4t+1 \right)}^{2}}} \\ & =\frac{-16{{t}^{\frac{1}{3}}}+{}^{2}/{}_{{{t}^{\tfrac{2}{3}}}}}{{{\left( 4t+1 \right)}^{2}}} \end{align*}\] <p>Note that we simplified the numerator more than usual here. This was only done to make the derivative easier to evaluate.</p> <p>The rate of change of the volume at $t = 8$ is then,</p> \[\begin{align*}V'\left( 8 \right) & = \frac{{ - 16\left( 2 \right) + \frac{2}{4}}}{{{{\left( {33} \right)}^2}}}\hspace{0.25in}\hspace{0.25in}{\left( 8 \right)^{\frac{1}{3}}} = 2\hspace{0.25in}{\left( 8 \right)^{\frac{2}{3}}} = {\left( {{{\left( 8 \right)}^{\frac{1}{3}}}} \right)^2} = {\left( 2 \right)^2} = 4\\ & = - \frac{{63}}{{2178}} = - \frac{7}{{242}}\end{align*}\] <p>So, the rate of change of the volume at $t = 8$ is negative and so the volume must be decreasing. Therefore, air is being drained out of the balloon at $t = 8$.</p> </div> </div> </div> <p>As a final topic let’s note that the product rule can be extended to more than two functions, for instance.</p> \[\begin{align*}{\left( {f\,g\,h} \right)^\prime } & = f'\,g\,h + f\,g'\,h + f\,g\,h'\\ & {\left( {f\,g\,h\,w} \right)^\prime } = f'\,g\,h\,w + f\,g'\,h\,w + f\,g\,h'\,w + f\,g\,h\,w'\end{align*}\] <p>Deriving these products of more than two functions is actually pretty simple. For example, let’s take a look at the three function product rule.</p> <p>First, we don’t think of it as a product of three functions but instead of the product rule of the two functions $f\,g$ and $h$ which we can then use the two function product rule on. Doing this gives,</p> \[{\left( {f\,g\,h} \right)^\prime } = {\left( {\left[ {f\,g} \right]\,h} \right)^\prime } = {\left[ {f\,g} \right]^\prime }h + \left[ {f\,g} \right]h'\] <p>Note that we put brackets on the $f\,g$ part to make it clear we are thinking of that term as a single function. Now all we need to do is use the two function product rule on the ${\left[ {f\,g} \right]^\prime }$ term and then do a little simplification.</p> \[{\left( {f\,g\,h} \right)^\prime } = \left[ {f'\,g + f\,g'} \right]h + \left[ {f\,g} \right]h' = f'\,g\,h + f\,g'\,h + f\,g\,h'\] <p>Any product rule with more functions can be derived in a similar fashion.</p> <p>With this section and the previous section we are now able to differentiate powers of $x$ as well as sums, differences, products and quotients of these kinds of functions. However, there are many more functions out there in the world that are not in this form. The next few sections give many of these functions as well as give their derivatives.</p> </div>  <div class="footer"> <div class="footer-links"> [<a href="/Contact.aspx">Contact Me</a>] [<a href="/Privacy.aspx">Privacy Statement</a>] [<a href="/Help.aspx">Site Help & FAQ</a>] [<a href="/Terms.aspx">Terms of Use</a>] </div> <div class="footer-dates"> <div class="footer-copyright"><span id="lblCopyRight">© 2003 - 2024 Paul Dawkins</span></div> <div class="footer-spacer"></div> <div class="footer-modified-date">Page Last Modified : <span id="lblModified">11/16/2022</span></div> </div> </div> </div>  </body> </html>

CINXE.COM

Calculus I - Product and Quotient Rule