{"id":170,"date":"2019-08-27T16:24:41","date_gmt":"2019-08-27T16:24:41","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=170"},"modified":"2020-06-17T18:56:43","modified_gmt":"2020-06-17T18:56:43","slug":"differentiating-special-functions","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/single-variable-calculus\/differentiating-special-functions\/","title":{"rendered":"Differentiating Special Functions"},"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<script src=\"https:\/\/www.math.hmc.edu\/jsMath\/easy\/load-dollars.js\"><\/script>\n\n\n\n<title>Differentiating Special Functions &#8211; HMC Calculus Tutorial<\/title>\n\n\n\n<p>\n<!------------------------>\n\nIn this tutorial, we review the differentiation of trigonometric,\nlogarithmic, and exponential functions.\n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Trigonometric Functions<\/h4>\n\n\n\n<p> The derivatives of the basic trigonometric functions are given here for reference. \\[\\begin{array}{rr} f(x) &amp; f'(x)\\\\ ~\\\\ \\qquad\\qquad\\sin x  &amp; \\cos x\\\\ \\cos x &amp; -\\sin x\\\\ \\tan x &amp; \\sec^2 x\\\\ \\sec x &amp; \\sec x \\tan x\\\\ \\csc x &amp; \\qquad\\qquad -\\csc x\\cot x\\\\ \\cot x &amp; -\\csc^2 x\\\\ ~\\\\ \\end{array}\\] The derivatives of $\\sin x$ and $\\cos x$ can be derived using the limit definition of the derivative.<\/p>\n\n\n<style>.kt-accordion-id_6a1a75-b2 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:0px;}.kt-accordion-id_6a1a75-b2 .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_6a1a75-b2 > .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_6a1a75-b2: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_6a1a75-b2: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_6a1a75-b2: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_6a1a75-b2: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_6a1a75-b2: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_6a1a75-b2 > .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_6a1a75-b2 .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_6a1a75-b2: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_6a1a75-b2: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_6a1a75-b2: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_6a1a75-b2: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_6a1a75-b2: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_6a1a75-b2: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_6a1a75-b2: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_6a1a75-b2: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_6a1a75-b2: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_6a1a75-b2: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_6a1a75-b2 .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id_6a1a75-b2 > .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_6a1a75-b2: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_6a1a75-b2: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_6a1a75-b2: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_6a1a75-b2: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_6a1a75-b2: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_6a1a75-b2 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id_6a1a75-b2 .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_6a1a75-b2 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_8333fa-cc\"><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\">Limit Definition of the Derivative<\/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>Limit Definition of the Derivative<\/title>\n<center><font size=\"+2\">\nLimit Definition of the Derivative\n<\/font><\/center>\n\n\n\n<p>\n\n\\[f'(x)=\\lim_{\\Delta x\\to 0}\\, \\frac{f(x+\\Delta x)-f(x)}{\\Delta x}\\] \n\n<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p style=\"text-align:left\">For $\\sin x$, \\begin{eqnarray*} \\frac{d}{dx}(\\sin x)&amp;=&amp;\\lim_{h\\to 0} \\frac{\\sin (x+h)-\\sin (x)}{h}\\\\ &amp;=&amp; \\lim_{h\\to 0} \\frac{(\\sin x\\cos h+\\cos x\\sin h)-\\sin x}{h}\\\\ &amp;=&amp; \\lim_{h\\to 0} \\left[\\sin x\\frac{\\cos h -1}{h}+\\cos x\\frac{\\sin h}{h}\\right]\\\\  &amp;=&amp;\\sin x\\lim_{h\\to 0}\\left[\\frac{\\cos h-1}{h}\\right]+\\cos x\\lim_{h\\to 0}\\left[\\frac{\\sin h}{h}\\right]\\\\ &amp;=&amp; \\sin x (0)+\\cos x (1)\\\\ &amp;=&amp; \\cos x. \\end{eqnarray*} The derivative of $\\cos x$ is derived analogously. Then the remaining derivatives can be derived using the quotient rule, since all the other trigonometric functions are quotients involving $\\sin x$ and $\\cos x$. <\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p> The derivative of $\\tan (x^2)$ is $\\displaystyle \\sec^2(x^2)\\cdot\\frac{d}{dx}(x^2) =2x\\sec^2(x^2)$ by the <b><a href=\"https:\/\/math.hmc.edu\/calculus-tutorials\/hmc-mathematics-calculus-online-tutorials\/single-variable-calculus\/chain-rule\/\">chain rule<\/a><\/b>. <\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Logarithmic Functions<\/h4>\n\n\n\n<p>\n\nBy the definition of the natural logarithm, $\\displaystyle\n\\frac{d}{dx}[\\ln x] =\\frac{1}{x}$ for $x&gt;0$.  Also, $\\displaystyle\n\\frac{d}{dx} [\\ln |x|]=\\frac{1}{x}$ for all $x\\neq 0$.  To see this,\nsuppose $x&lt;0$.  Then $\\ln |x|=\\ln (-x)$.  \n\n<\/p>\n\n\n\n<p>\nSo \n\\begin{eqnarray*}\n\t\\frac{d}{dx}[\\ln |x|]&amp;=&amp;\\frac{d}{dx}[\\ln (-x)]\\\\\n\t&amp;=&amp; \\frac{d}{dx}(-x)\\left(\\frac{1}{-x}\\right)\\\\\n\t&amp;=&amp; (-1)\\left(\\frac{1}{-x}\\right)\\\\\n\t&amp;=&amp; \\frac{1}{x}.\n\\end{eqnarray*}\n\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p> By the <b><a href=\"https:\/\/math.hmc.edu\/calculus-tutorials\/hmc-mathematics-calculus-online-tutorials\/single-variable-calculus\/chain-rule\/\">chain rule<\/a><\/b>, the derivative of  $\\ln (x^3+5)$ is $\\displaystyle \\frac{d(x^3+5)}{dx}\\cdot\\frac{1}{x^3+5}=\\frac{3x^2}{x^3+5}$. <\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Exponential Functions<\/h4>\n\n\n\n<p>\n\nThere is an elegant way to show that $\\displaystyle\n\\frac{d}{dx}\\left[e^x\\right]=e^x$.  \nWe start with the identity $\\displaystyle \\ln (e^x)=x$.\nDifferentiating both sides,\n\\begin{eqnarray*}\n\t\\frac{d}{dx}[\\ln (e^x)]&amp;=&amp;\\frac{d}{dx}(x)\\\\\n\t\\frac{d}{dx}[\\ln (e^x)]&amp;=&amp; 1\\\\\n\t\\frac{d}{dx}(e^x)\\cdot \\frac{1}{e^x}&amp;=&amp;1\\\\\n\t\\frac{d}{dx}(e^x)&amp;=&amp;e^x.\n\\end{eqnarray*}\nSince $e^x$ is never $0$, this derivation holds for all $x$.\n\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\nThe derivative of $\\displaystyle e^{-3x+2}$ is $\\displaystyle\ne^{-3x+2}\\cdot \\frac{d}{dx}(-3x+2)=-3e^{-3x+2}$.\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>\n\n\\[\\begin{array}{rr}\n\tf(x) &amp; f'(x)\\\\\n\t~\\\\\n\t\\qquad\\qquad\\sin x  &amp; \\cos x\\\\\n\t\\cos x &amp; -\\sin x\\\\\n\t\\tan x &amp; \\sec^2 x\\\\\n\t\\sec x &amp; \\sec x \\tan x\\\\\n\t\\csc x &amp; \\qquad\\qquad -\\csc x\\cot x\\\\\n\t\\cot x &amp; -\\csc^2 x\\\\\n\t\\ln x &amp; \\frac{1}{x}\\\\\n\te^x &amp; e^x\\\\\n\t~\\\\\n\\end{array}\\]\n\n<br><\/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\/QZ0610\/\">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>Differentiating Special Functions &#8211; HMC Calculus Tutorial In this tutorial, we review the differentiation of trigonometric, logarithmic, and exponential functions. Trigonometric Functions The derivatives of the basic trigonometric functions are given here for reference. \\[\\begin{array}{rr} f(x) &amp; f'(x)\\\\ ~\\\\ \\qquad\\qquad\\sin x &amp; \\cos x\\\\ \\cos x &amp; -\\sin x\\\\ \\tan x &amp; \\sec^2 x\\\\ \\sec&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-170","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/170","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=170"}],"version-history":[{"count":5,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/170\/revisions"}],"predecessor-version":[{"id":1087,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/170\/revisions\/1087"}],"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=170"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=170"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}