{"id":166,"date":"2019-08-22T20:05:56","date_gmt":"2019-08-22T20:05:56","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=166"},"modified":"2019-12-02T05:58:02","modified_gmt":"2019-12-02T05:58:02","slug":"chain-rule","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/single-variable-calculus\/chain-rule\/","title":{"rendered":"Chain 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<p>You probably remember the derivatives of $\\sin (x)$, $x^{8}$, and $e^{x}$.  But what about functions like $\\sin (2x-1)$, $(3x^{2}-4x+1)^{8}$, or $e^{-x^{2}}$?  How do we take the derivative of compositions of functions? <\/p>\n\n\n<style>.kt-accordion-id_2f748b-d5 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_2f748b-d5 .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_2f748b-d5 > .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_2f748b-d5: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_2f748b-d5: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_2f748b-d5: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_2f748b-d5: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_2f748b-d5: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_2f748b-d5 > .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_2f748b-d5 .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_2f748b-d5: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_2f748b-d5: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_2f748b-d5: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_2f748b-d5: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_2f748b-d5: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_2f748b-d5: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_2f748b-d5: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_2f748b-d5: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_2f748b-d5: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_2f748b-d5: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_2f748b-d5 .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_2f748b-d5 > .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_2f748b-d5: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_2f748b-d5: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_2f748b-d5: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_2f748b-d5: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_2f748b-d5: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_2f748b-d5 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_2f748b-d5 .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_2f748b-d5 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_8cab99-d6\"><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\">Compositions<\/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<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n<title>Compositions of Functions<\/title>\n<center><font size=\"+2\">\nCompositions of Functions\n<\/font><\/center>\n\n\n\n<p>\n\n The <strong>composition<\/strong> $f(g(x))$, often denoted by $f\\circ g$, of functions $f$ and $g$ is the function obtained by applying the function $f$ to $g(x)$.  \n\n<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p> <\/p>\n\n\n\n<p>The <strong>Chain Rule<\/strong>  allows us to use our knowledge of the derivatives of functions $f(x)$  and $g(x)$ to find the derivative of the composition $f(g(x))$: <\/p>\n\n\n\n<p>\n\tSuppose a function $g(x)$ is differentiable at $x$ and $f(x)$ is\n\tdifferentiable at $g(x)$.  Then the composition $f(g(x))$ is\n\tdifferentiable at $x$.\n\t<\/p>\n\n\n\n<p> Letting $y=f(g(x))$ and $u=g(x)$, $$ \\frac{dy}{dx}=\\frac{dy}{du}\\cdot\\frac{du}{dx}. $$ Using alternative notation, \\begin{eqnarray*} \\frac{d}{dx}\\left[ f(g(x)) \\right] &amp; = &amp; f'(g(x))g'(x), \\\\ \\frac{d}{dx}\\left[ f(u) \\right] &amp; = &amp; f'(u)\\frac{du}{dx}.  \\end{eqnarray*} <\/p>\n\n\n<style>.kt-accordion-id_5cf7f3-d0 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_5cf7f3-d0 .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_5cf7f3-d0 > .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_5cf7f3-d0: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_5cf7f3-d0: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_5cf7f3-d0: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_5cf7f3-d0: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_5cf7f3-d0: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_5cf7f3-d0 > .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_5cf7f3-d0 .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_5cf7f3-d0: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_5cf7f3-d0: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_5cf7f3-d0: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_5cf7f3-d0: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_5cf7f3-d0: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_5cf7f3-d0: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_5cf7f3-d0: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_5cf7f3-d0: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_5cf7f3-d0: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_5cf7f3-d0: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_5cf7f3-d0 .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_5cf7f3-d0 > .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_5cf7f3-d0: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_5cf7f3-d0: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_5cf7f3-d0: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_5cf7f3-d0: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_5cf7f3-d0: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_5cf7f3-d0 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_5cf7f3-d0 .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_5cf7f3-d0 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_0f29a1-d7\"><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 the Chain 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<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n<title>Proof of the Chain Rule<\/title>\n<center><font size=\"+2\">\nProof of the Chain Rule\n<\/font><\/center>\n\n\n\n<p>   Let $g(x)$ be differentiable at $x$ and $f(x)$ be differentiable at $g(x)$.  Let $y = f(g(x))$ and $u=g(x)$.  <\/p>\n\n\n\n<p>\n\nWe will use the fact that if $y=h(x)$ is differentiable at $x$ then $$ \\Delta y = h'(x) \\Delta x + \\varepsilon \\Delta x $$ where $\\varepsilon \\rightarrow 0$ as $\\Delta x \\rightarrow 0$.  We have that \\begin{eqnarray*} \\Delta u &amp; = &amp; g'(x) \\Delta x + \\varepsilon_1 \\Delta x \\mbox{ where } \\varepsilon_1 \\rightarrow 0 \\mbox{ as } \\Delta x \\rightarrow 0,\\\\ \\Delta y &amp; = &amp; f'(u) \\Delta u + \\varepsilon_2 \\Delta u  \\mbox{ where } \\varepsilon_2 \\rightarrow 0 \\mbox{ as } \\Delta u \\rightarrow 0. \\end{eqnarray*} Substituting $\\Delta u$ from the first equation into the second, $$ \\frac{\\Delta y}{\\Delta x} = \\left[ f'(u) + \\varepsilon_2 \\right] \\left[ g'(x) + \\varepsilon_1 \\right]. $$ Taking the limit as $\\Delta x \\rightarrow 0$, \\begin{eqnarray*} \\frac{dy}{dx} &amp; = &amp; f'(u) \\cdot g'(x) \\\\  &amp; = &amp; \\frac{dy}{du} \\cdot \\frac{du}{dx} \\end{eqnarray*} (taken from <em>Calculus<\/em>, by Howard Anton.)  \n\n<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p>  <\/p>\n\n\n\n<p>The three formulations of the Chain Rule given here are identical in meaning.  In words, the derivative of $f(g(x))$ is the derivative of $f$, evaluated at $g(x)$, multiplied by the derivative of $g(x)$. <\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Examples<\/h6>\n\n\n\n<ul class=\"wp-block-list\"><li> To differentiate $\\sin (2x-1)$, we identify $u=2x-1$.  Then\n<p align=\"center\">\n<\/p><p align=\"center\">\n\t\t\t\\begin{eqnarray*}\n\t\t\t\\frac{d}{dx}\\left[ \\sin(2x-1)\\right] &amp; = &amp; \\frac{d}{du} \\left[ \\sin\n\t\t\t (u) \\right] \\cdot \\frac{d}{dx} \\left[ 2x-1 \\right] \\\\\n\t\t\t &amp; = &amp; \\cos (u) \\cdot 2 \\\\\n\t\t\t &amp; = &amp; 2 \\cos (2x-1). \n\t\t\t\\end{eqnarray*}\n<\/p>\n<p align=\"center\">\n<\/p><p align=\"center\">\n\t\t\t\t\\begin{eqnarray*}\n\t\t\t\tf(x) &amp; = &amp; \\sin (x) \\\\\n\t\t\t\tg(x) &amp; = &amp; 2x-1 \\\\\n\t\t\t\tf(g(x)) &amp; = &amp; \\sin (2x-1)\n\t\t\t\t\\end{eqnarray*}\n<\/p>\n<p><\/p>\n<p><\/p>\n\n<p>\n<\/p><\/li><li> To differentiate $\\left( 3x^{2} &#8211; 4x + 1 \\right)^{8}$, we\n\tidentify $u=3x^2-4x+1$.  Then\n<p align=\"center\">\n<\/p><p align=\"center\">\n\t\t\t\\begin{eqnarray*}\n\t\t\t\\frac{d}{dx} \\left[\\left( 3x^{2} &#8211; 4x + 1 \\right)^{8}\\right] &amp; = &amp;\n\t\t\t\\frac{d}{du}\\left[ u^8 \\right] \\cdot \\frac{d}{dx}\\left[ 3x^2-4x+1\n\t\t\t\\right] \\\\\n\t\t\t &amp; = &amp; 8u^7 \\cdot (6x-4) \\\\\n\t\t\t &amp; = &amp; 8 (6x-4)\\left( 3x^{2} &#8211; 4x + 1 \\right)^{7}.\n\t\t\t\\end{eqnarray*}\n<\/p>\n<p align=\"center\">\n<\/p><p align=\"center\">\n\t\t\t\t\\begin{eqnarray*}\n\t\t\t\tf(x) &amp; = &amp; x^8 \\\\\n\t\t\t\tg(x) &amp; = &amp; 3x^{2} &#8211; 4x + 1 \\\\\n\t\t\t\tf(g(x)) &amp; = &amp; \\left( 3x^{2} &#8211; 4x + 1 \\right)^{8}\n\t\t\t\t\\end{eqnarray*}\n<\/p>\n<p><\/p>\n<p><\/p>\n\n<p>\n<\/p><\/li><li> To differentiate $e^{-x^{2}}$, we identify $u=-x^{2}$.  Then\n<p align=\"center\">\n<\/p><p align=\"center\">\n\t\t\t\\begin{eqnarray*}\n\t\t\t\\frac{d}{dx} \\left[ e^{-x^{2}} \\right] &amp; = &amp; \\frac{d}{du}\\left[ e^u\n\t\t\t\\right] \\cdot \\frac{d}{dx}\\left[ -x^2 \\right] \\\\\n\t\t\t&amp; = &amp; e^u \\cdot (-2x) \\\\\n\t\t\t&amp; = &amp; -2xe^{-x^2}.\n\t\t\t\\end{eqnarray*}\n<\/p>\n<p align=\"center\">\n<\/p><p align=\"center\">\n\t\t\t\t\\begin{eqnarray*}\n\t\t\t\tf(x) &amp; = &amp; e^x \\\\\n\t\t\t\tg(x) &amp; = &amp; -x^2 \\\\\n\t\t\t\tf(g(x)) &amp; = &amp; e^{-x^{2}}\n\t\t\t\t\\end{eqnarray*}\n<\/p>\n<p><\/p>\n<p><\/p>\n\n<\/li><\/ul>\n\n\n\n<p>\nSometimes you will need to apply the Chain Rule several times in order\nto differentiate a function.\n\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\nWe will differentiate $\\sqrt{\\sin^{2} (3x) + x}$.\n\n$$\n\\begin{array}{rclc}\n\t\\frac{d}{dx}\\left[ \\sqrt{\\sin^{2} (3x) + x}\\right] &amp; = &amp;\n\t\\frac{1}{2\\sqrt{\\sin^{2} (3x) + x}} \\cdot \\frac{d}{dx}\\left[ \\sin^{2} (3x) + x\n\t\\right] &amp; f(u) = \\sqrt{u}\\\\\n\t&amp; = &amp; \\frac{1}{2\\sqrt{\\sin^{2} (3x) + x}} \\cdot \\left( 2 \\sin (3x)\n\t\\frac{d}{dx} \\left[ \\sin (3x) \\right] + 1 \\right) &amp; \n\t\\begin{array}{rcl}\n\t\tf(u) &amp; = &amp; u^2 \\\\\n\t\t\\frac{d}{dx} [x] &amp; = &amp; 1 \n\t\\end{array} \\\\\n\t&amp; = &amp; \\frac{1}{2\\sqrt{\\sin^{2} (3x) + x}} \\cdot \\left( 2 \\sin (3x)\n\t\\cos (3x) \\frac{d}{dx} [3x] + 1 \\right) &amp; f(u) = \\sin (u) \\\\\n\t&amp; = &amp; \\frac{1}{2\\sqrt{\\sin^{2} (3x) + x}} \\cdot \\left( 2 \\sin (3x)\n\t\\cos (3x) \\cdot 3 + 1 \\right) \\\\\n\t&amp; = &amp; \\displaystyle\\frac{6 \\sin (3x) \\cos (3x) + 1}{2\\sqrt{\\sin^{2} (3x) + x}}\n\\end{array}\n$$\n\n<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<center>\n<h4>Key Concepts<\/h4>\n<\/center>\n\n\n\n<p>\nLet $g(x)$ be differentiable at $x$ and $f(x)$ be differentiable at\n$f(g(x))$.  Then, if $y=f(g(x))$ and $u=g(x)$,\n$$\n\\frac{dy}{dx} = \\frac{dy}{du}\\cdot\\frac{du}{dx}.\n$$\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\/QZ3510\/\">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>You probably remember the derivatives of $\\sin (x)$, $x^{8}$, and $e^{x}$. But what about functions like $\\sin (2x-1)$, $(3x^{2}-4x+1)^{8}$, or $e^{-x^{2}}$? How do we take the derivative of compositions of functions? The Chain Rule allows us to use our knowledge of the derivatives of functions $f(x)$ and $g(x)$ to find the derivative of the composition&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-166","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/166","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=166"}],"version-history":[{"count":12,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/166\/revisions"}],"predecessor-version":[{"id":1080,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/166\/revisions\/1080"}],"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=166"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=166"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}