{"id":188,"date":"2019-08-27T17:12:28","date_gmt":"2019-08-27T17:12:28","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=188"},"modified":"2019-12-02T21:19:36","modified_gmt":"2019-12-02T21:19:36","slug":"product-rule","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/single-variable-calculus\/product-rule\/","title":{"rendered":"Product Rule"},"content":{"rendered":"\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n\n\n\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n\n\n\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=EmulateIE7\">\n\n\n\n<title>Product Rule for Derivatives &#8211; HMC Calculus Tutorial<\/title>\n\n\n\n<p>\n<!------------------------>\n\nIn Calculus and its applications we often encounter functions that are\nexpressed as the product of two other functions, like the following examples:\n<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li> $h(x) = x e^x = (x)(e^x),$\n\t<br><br>\n\t<\/li><li> $h(x) = x^2 \\sin x = (x^2)(\\sin x),$\n\t<br><br>\n\t<\/li><li> $h(x) = e^{-x^2} \\cos 2x = (e^{-x^2})(\\cos 2x).$\n<\/li><\/ul>\n\n\n\n<p> In each of these examples, the values of the function $h$ can be written in the form  $$   h(x) = f(x) g(x) $$ for functions $f(x)$ and $g(x)$. If we know the derivative of $f(x)$ and  $g(x)$, the <strong>Product Rule<\/strong>  provides a formula for the derivative of  $h(x) = f(x) g(x)$:   <\/p>\n\n\n\n<center>\n<p align=\"center\">\n\t\t$h'(x) = \\left[f(x)g(x)\\right]&#8217; = f'(x) g(x) + f(x) g'(x).$\n<\/p>\n<\/center>\n\n\n<style>.kt-accordion-id_a53996-58 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_a53996-58 .kt-accordion-panel-inner{border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 1.5rem);}.kt-accordion-id_a53996-58 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header{border-top-color:#555555;border-right-color:#555555;border-bottom-color:#555555;border-left-color:#555555;border-top-width:0px;border-right-width:0px;border-bottom-width:0px;border-left-width:0px;border-top-left-radius:0px;border-top-right-radius:0px;border-bottom-right-radius:0px;border-bottom-left-radius:0px;background:#f2f2f2;font-size:18px;line-height:24px;color:#555555;padding-top:10px;padding-right:14px;padding-bottom:10px;padding-left:14px;}.kt-accordion-id_a53996-58:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_a53996-58:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap .kt-blocks-accordion-icon-trigger:before{background:#555555;}.kt-accordion-id_a53996-58:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger{background:#555555;}.kt-accordion-id_a53996-58:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_a53996-58:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-icon-trigger:before{background:#f2f2f2;}.kt-accordion-id_a53996-58 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header:hover, \n\t\t\t\tbody:not(.hide-focus-outline) .kt-accordion-id_a53996-58 .kt-blocks-accordion-header:focus-visible{color:#444444;background:#eeeeee;border-top-color:#eeeeee;border-right-color:#eeeeee;border-bottom-color:#eeeeee;border-left-color:#eeeeee;}.kt-accordion-id_a53996-58:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_a53996-58:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_a53996-58:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion--visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_a53996-58:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle ) .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#444444;}.kt-accordion-id_a53996-58:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger, body:not(.hide-focus-outline) .kt-accordion-id_a53996-58:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger{background:#444444;}.kt-accordion-id_a53996-58:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_a53996-58:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:hover .kt-blocks-accordion-icon-trigger:before, body:not(.hide-focus-outline) .kt-accordion-id_a53996-58:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:after, body:not(.hide-focus-outline) .kt-accordion-id_a53996-58:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible .kt-blocks-accordion-icon-trigger:before{background:#eeeeee;}.kt-accordion-id_a53996-58 .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_a53996-58 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{color:#ffffff;background:#444444;border-top-color:#444444;border-right-color:#444444;border-bottom-color:#444444;border-left-color:#444444;}.kt-accordion-id_a53996-58:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_a53996-58:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id_a53996-58:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger{background:#ffffff;}.kt-accordion-id_a53996-58:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id_a53996-58:not( .kt-accodion-icon-style-basic ):not( .kt-accodion-icon-style-xclose ):not( .kt-accodion-icon-style-arrow ) .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:before{background:#444444;}@media all and (max-width: 767px){.kt-accordion-id_a53996-58 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_a53996-58 .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:0px;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-wrap kt-accordion-id_a53996-58 kt-accordion-has-2-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"true\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane_63897c-07\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><div class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Proof of Product Rule<\/span><\/div><div class=\"kt-blocks-accordion-icon-trigger\"><\/div><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p>\n\n The derivative $h'(x)$ is given by the limit formula: $$     h'(x) = \\lim_{\\Delta x \\to 0} \\frac{h(x + \\Delta x) &#8211; h(x)}{\\Delta x},   $$ provided the limit exists. We now express $h$ using the product of  $f$ and $g$,     \\begin{eqnarray*}    h'(x) &amp; = &amp; \\lim_{\\Delta x \\to 0} \\frac{h(x + \\Delta x) &#8211; h(x)}{\\Delta x}\\\\          &amp; = &amp; \\lim_{\\Delta x \\to 0} \\frac{f(x + \\Delta x)g(x + \\Delta x) &#8211; f(x)g(x)}{\\Delta x}. \\end{eqnarray*} To make further progress, we need to relate our limit formula to the limit formulas for the derivatives $f'(x)$ and $g'(x)$, namely   $$ f'(x) = \\lim_{\\Delta x \\to 0} \\frac{f(x + \\Delta x) &#8211; f(x)}{\\Delta x}  \\quad \\mbox{and} \\quad  g'(x) = \\lim_{\\Delta x \\to 0} \\frac{g(x + \\Delta x) &#8211; g(x)}{\\Delta x}. $$ To relate these formulas to the limit for $h'(x)$, we use the &#8220;trick&#8221;  of adding and subtracting the term $f(x)g(x + \\Delta x)$ in the numerator, and then simplifying: \\begin{eqnarray*} h'(x) &amp; = &amp; \\lim_{\\Delta x \\to 0} \\frac{f(x + \\Delta x)g(x + \\Delta x) &#8211;  f(x)g(x)}{\\Delta x}\\\\   &amp; = &amp; \\lim_{\\Delta x \\to 0} \\frac{f(x + \\Delta x)g(x + \\Delta x)    + (f(x)g(x + \\Delta x) &#8211; f(x)g(x + \\Delta x)) &#8211; f(x)g(x)}{\\Delta x}\\\\   &amp;=&amp; \\lim_{\\Delta x \\to 0} \\left[\\left(\\frac{f(x + \\Delta x) &#8211; f(x)}{\\Delta x}\\right) g(x + \\Delta x) +    \\left( \\frac{g(x + \\Delta x) &#8211; g(x)}{\\Delta x} \\right) f(x) \\right]\\\\   &amp; = &amp; \\lim_{\\Delta x \\to 0} \\frac{f(x + \\Delta x) &#8211; f(x)}{\\Delta x}    \\lim_{\\Delta x \\to 0} g(x + \\Delta x) +   f(x) \\lim_{\\Delta x \\to 0} \\frac{g(x + \\Delta x) &#8211; g(x)}{\\Delta x}\\\\   &amp; = &amp; f'(x) g(x) + f(x) g'(x). \\end{eqnarray*} The last two steps are justified by assuming that $f'(x)$ and $g'(x)$ exist.  \n\n<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p><\/p>\n\n\n\n<p>\nWe illustrate this rule with the following examples.\n<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li> If $h(x) = x e^x $ then \n\t\t\\begin{eqnarray*}\n\t\t\th'(x) &amp;=&amp; (x)&#8217; e^x + x (e^x)&#8217;\\\\ \n\t\t\t       &amp;=&amp;  e^x + xe^x. \n\t\t\\end{eqnarray*}\n\t<br><br>\n\t<\/li><li> If $h(x) = x^2 \\sin x $ then \n\t\t\\begin{eqnarray*}\n\t\t\th'(x) &amp;=&amp; (x^2)&#8217; \\sin x + (x^2)(\\sin x)&#8217;\\\\\n\t\t\t      &amp;=&amp;  2x \\sin x + x^2 \\cos x. \n\t\t\\end{eqnarray*}\n\t<br><br>\n\t<\/li><li> If $h(x) = e^{-x^2} \\cos 2x $ then \n\t\t\\begin{eqnarray*}\n\t\t\th'(x) &amp;=&amp; (e^{-x^2})&#8217; \\cos 2x + e^{-x^2} (\\cos 2x)&#8217; \\\\\n\t\t\t      &amp;=&amp; -2xe^{-x^2} \\cos 2x -2e^{-x^2} \\sin 2x.\n\t\t\\end{eqnarray*} \n<\/li><\/ul>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<center>\n<p>\n<\/p><h4>Key Concepts<\/h4>\n<p>\n<\/p><\/center>\n\n\n\n<b>Product Rule<\/b>\n\n\n\n<p>\nLet $f(x)$ and $g(x)$ be differentiable at $x$. Then $h(x) = f(x)g(x)$ is \ndifferentiable at $x$ and\n$h'(x) = f'(x)g(x) + f(x)g'(x)$.\n\n<!------------------------>\n\n\n<br>\n\n<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>\n\n[<a href=\"https:\/\/physics.hmc.edu\/ct\/quiz\/QZ4210\/\">I&#8217;m ready to take the quiz.<\/a>]\n[<a href=\"#top\">I need to review more.<\/a>]<br>\n\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Product Rule for Derivatives &#8211; HMC Calculus Tutorial In Calculus and its applications we often encounter functions that are expressed as the product of two other functions, like the following examples: $h(x) = x e^x = (x)(e^x),$ $h(x) = x^2 \\sin x = (x^2)(\\sin x),$ $h(x) = e^{-x^2} \\cos 2x = (e^{-x^2})(\\cos 2x).$ In each&hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":57,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-188","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/188","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/comments?post=188"}],"version-history":[{"count":5,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/188\/revisions"}],"predecessor-version":[{"id":1094,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/188\/revisions\/1094"}],"up":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/57"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/media?parent=188"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=188"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}