{"id":37829,"date":"2018-02-13T21:31:04","date_gmt":"2018-02-13T21:31:04","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=37829"},"modified":"2026-03-25T12:48:44","modified_gmt":"2026-03-25T17:48:44","slug":"product-and-quotient-rule","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/product-and-quotient-rule\/","title":{"rendered":"Product and Quotient Rules in Calculus"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_iq1xF28yvGk\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_iq1xF28yvGk\" data-source-videoID=\"iq1xF28yvGk\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Product and Quotient Rules in Calculus Video\" height=\"720\" width=\"1280\" class=\"size-full\" data-matomo-title = \"Product and Quotient Rules in Calculus\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_iq1xF28yvGk:hover {cursor:pointer;} img#videoThumbnailImage_iq1xF28yvGk {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/10\/thumb1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_iq1xF28yvGk\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_iq1xF28yvGk\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Product and Quotient Rules in Calculus\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_iq1xF28yvGk\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_iq1xF28yvGk\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_iq1xF28yvGk\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction zVe_Function() {\n  var x = document.getElementById(\"zVe\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"zVe_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"zVe\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#The_Product_Rule\" class=\"smooth-scroll\">The Product Rule<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Product_Rule_Examples\" class=\"smooth-scroll\">Product Rule Examples<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#The_Quotient_Rules\" class=\"smooth-scroll\">The Quotient Rules<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Quotient_Rule_Examples\" class=\"smooth-scroll\">Quotient Rule Examples<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>You may have seen how to take the derivative of constants, powers of \\(x\\), and polynomials, but how would you take the derivative of something like this?<\/p>\n<div class=\"examplesentence\">\\(f(x)=\\frac{6x}{2x^{2}+5}+x\\text{ }sin(x)\\)<\/div>\n<p>\n&nbsp;<\/p>\n<p>In this video, I\u2019m going to show you how to take the derivative when a given function includes the multiplication or division of two or more differentiable functions.<\/p>\n<p>First, let\u2019s talk about how to handle derivatives involving the multiplication of functions. To take such derivatives, we\u2019ll need something called the product rule.<\/p>\n<h2><span id=\"The_Product_Rule\" class=\"m-toc-anchor\"><\/span>The Product Rule<\/h2>\n<p>\nThe <strong>product rule<\/strong> states that if you want to take the derivative of two functions multiplied together, start by writing the first function and multiplying it by the derivative of the second function. Then, add the second function multiplied by the derivative of the first function.<\/p>\n<p>Mathematicians write the product rule in this generalized form:<\/p>\n<div class=\"examplesentence\">\\(\\frac{d}{dx}(fg)=fg&#8217;+gf&#8217;\\)<\/div>\n<p>\n&nbsp;<\/p>\n<p>In this case, \\(f\\) and \\(g\\) are two things being multiplied together, and \\(f&#8217;\\) and \\(g&#8217;\\) are their respective derivatives.<\/p>\n<h2><span id=\"Product_Rule_Examples\" class=\"m-toc-anchor\"><\/span>Product Rule Examples<\/h2>\n<h3><span id=\"Example_1\" class=\"m-toc-anchor\"><\/span>Example 1<\/h3>\n<p>\nLet\u2019s try some examples of using the product rule. Let\u2019s take a look at the function:<\/p>\n<div class=\"examplesentence\">\\(p(x)=x^{4} ln(x)\\)<\/div>\n<p>\n&nbsp;<br \/>\nTo find the derivative of this function, remember that the product rule says that the derivative of (fg)=fg&#8217;+gf&#8217;[\/latex]. In this case, \\(f\\) is equal to the first function, \\(x^{4}\\), and \\(g\\) is equal to the second function, which is \\(ln(x)\\).<\/p>\n<p>We need to write down \\(fg&#8217;\\). In this case, that will be \\(x^\\times \\frac{1}{x}\\), since the derivative of \\(ln(x)\\) is \\(\\frac{1}{x}\\). <\/p>\n<div class=\"examplesentence\">\\(fg&#8217;=x^{4}\\cdot \\frac{1}{x}\\)<\/div>\n<p>\n&nbsp;<br \/>\nTogether, these can just be reduced to \\(x^{3}\\).<\/p>\n<div class=\"examplesentence\">\\(fg&#8217;=x^{3}\\)<\/div>\n<p>\n&nbsp;<br \/>\nThe last step in using the product rule is writing the term \\(gf&#8217;\\). Remember, in this problem \\(g=ln(x)\\), so we will write that multiplied by the derivative of \\(f\\). Since the derivative of \\(x^{4}\\) is \\(4x^{3}\\), we write \\(ln(x)\\cdot 4x^{3}\\). The derivative of the function \\(p(x)=x^{4}ln(x)\\) is then equal to<\/p>\n<div class=\"examplesentence\">\\(gf&#8217;=x^{3}+ln(x)\\cdot 4x^{3}\\)<\/div>\n<p>\n&nbsp;<br \/>\nThe derivative of the function \\(p(x)=x^{4} ln(x)\\) is then equal to \\(gf&#8217;=x^{3}+ln(x)\\cdot 4x^{3}\\).<\/p>\n<h3><span id=\"Example_2\" class=\"m-toc-anchor\"><\/span>Example 2<\/h3>\n<p>\nLet\u2019s try another example. Use the product rule to find the derivative of \\(q(x)=e^xcos(x)\\). Remember, the product rule says that the derivative will equal the first function times the derivative of the second, plus the second function times the derivative of the first.<\/p>\n<p>Let\u2019s write down the first function, \\(e^{x}\\), multiplied by the derivative of the second function. Since the derivative of \\(cos(x)\\) is \\(-sin(x)\\), we get:<\/p>\n<div class=\"examplesentence\">\\(fg&#8217;=e^{x}[-sin(x)]\\)<\/div>\n<p>\n&nbsp;<br \/>\nThen, we are going to add the second function multiplied by the derivative of the first function. Since \\(e^{x}\\) is its own derivative, \\(gf&#8217;=cos(x)\\cdot e^x\\).<\/p>\n<p>Combining both parts, we get:<\/p>\n<div class=\"examplesentence\">\\(q'(x)=e^{x}-sin(x))+cos(x)e^{x}\\)<\/div>\n<p>\n&nbsp;<br \/>\nWe can clean this up a little bit by factoring the \\(e^{x}\\) out from both terms and rearranging a bit:<\/p>\n<div class=\"examplesentence\">\\(q'(x)=e^{x}[cos(x)-sin(x)]\\)<\/div>\n<p>\n&nbsp;<\/p>\n<h3><span id=\"Example_3\" class=\"m-toc-anchor\"><\/span>Example 3<\/h3>\n<p>\nLet\u2019s take a look at one more example. Use the product rule to find the derivative of \\(r(x)=2xe^{x}\\). <\/p>\n<p>Now this is technically three things multiplied together: 2, \\(x\\), and \\(e^{x}\\). But, we can combine 2 and \\(x\\) as \\(f\\) and leave \\(e^x\\) as \\(g\\). Using these definitions, the derivative of \\(r(x)\\) is equal to the first piece times the derivative of the second, which is \\(2x\\cdot e^{x}\\), plus the second piece times the derivative of the first, which is \\(e^x\\cdot 2\\).<\/p>\n<p>Simplifying, we get \\(2xe^{x}+2e^x\\), or we can factor out the \\(2e^{x}\\) and get \\(2e^{x}(x+1)\\).<\/p>\n<p>Some students have difficulty committing the product rule to memory. However, with some practice on your own, you should start to become comfortable with using it.<\/p>\n<h2><span id=\"The_Quotient_Rules\" class=\"m-toc-anchor\"><\/span>The Quotient Rules<\/h2>\n<p>\nWhile the product rule helps us find derivatives of functions containing multiplication, we have another tool called the quotient rule to help us find derivatives of functions containing division.<\/p>\n<p>The <strong>quotient rule<\/strong> states that the derivative of a fraction is a new fraction, whose numerator equals the original fraction\u2019s bottom times its top\u2019s derivative, minus the top times the bottom\u2019s derivative, and whose denominator equals the original fraction\u2019s bottom squared. <\/p>\n<p>If you didn\u2019t catch all that, don\u2019t worry. I have a way to help you remember the quotient rule.<\/p>\n<h3><span id=\"Remembering_the_Quotient_Rule\" class=\"m-toc-anchor\"><\/span>Remembering the Quotient Rule<\/h3>\n<p>\nLet\u2019s say we want to take the derivative of this fraction: \\(\\frac{\\text{High}}{\\text{Low}}\\).<\/p>\n<p>By \u201chigh\u201d I mean to indicate the numerator, and by \u201clow,\u201d the denominator. To get the derivative of this fraction, the quotient rule can be remembered in this way:<\/p>\n<div class=\"examplesentence\">\u201cLow d High, minus High d Low, over the square of what\u2019s below.\u201d<\/div>\n<p>\n&nbsp;<br \/>\nThis little saying is a memory tool to help you remember the quotient rule. <\/p>\n<p>Now here\u2019s what that means: The derivative of a fraction will be another fraction. The top of this fraction is given by the first part of the rhyme: \u201cLow d High, minus High d Low.\u201d Write the denominator\u2014\u201cLow\u201d\u2014and multiply by the derivative of the numerator\u2014that\u2019s what I mean by \u201cd High\u201d. Then, subtract the numerator\u2014\u201cHigh\u201d\u2014multiplied with the derivative of the denominator\u2014\u201cd Low.\u201d<\/p>\n<p>The second line of the rhyme says, \u201cover the square of what\u2019s below.\u201d This means that the denominator of the solution is equal to the starting denominator squared. <\/p>\n<p>To use the letters \\(f\\) and \\(g\\), the quotient rule can be written in this way:<\/p>\n<div class=\"examplesentence\">\\(\\frac{d}{dx}(\\frac{f}{g})=\\frac{gf&#8217;-fg&#8217;}{g^{2}}\\)<\/div>\n<p>\n&nbsp;<br \/>\nRemember: \u201cLow d High, minus High d Low, over the square of what\u2019s below.\u201d<\/p>\n<h2><span id=\"Quotient_Rule_Examples\" class=\"m-toc-anchor\"><\/span>Quotient Rule Examples<\/h2>\n<h3><span id=\"Example_1\" class=\"m-toc-anchor\"><\/span>Example 1<\/h3>\n<p>\nLet\u2019s try this out on an example. Use the quotient rule to find the derivative of the function \\(s(x)=\\frac{9x}{x^{3}-1}\\)<br \/>\n&nbsp;<br \/>\nSince this function is in fraction form, we know that its derivative will also be a fraction. The top of this fraction will equal \u201cLow d High minus High d Low.\u201d In this case, \u201cLow\u201d equals \\(x^{3}-1\\), and \u201cd High\u201d equals the derivative of \\(9x\\), which is 9. <\/p>\n<p>So the first thing we have is \\(s'(x)=(x^{3}-1)\\cdot 9\\). <\/p>\n<p>We are going to subtract \u201cHigh d Low,\u201d which is \\(9x\\) times the derivative of the bottom, which is \\(3x^{2}\\). So the numerator of the derivative will be equal to \\(s'(x)=(x^{3}-1)\\cdot 9-9x(3x^{2})\\). Let\u2019s multiply and collect like terms to clean this up a little bit.<\/p>\n<p>We can distribute our 9 into our parentheses and get:<\/p>\n<div class=\"examplesentence\">\\(s'(x)=9x^{3}-9-27x^{3}\\)<\/div>\n<p>\n&nbsp;<br \/>\nThen, we can combine our \\(x^{3}\\) terms and we&#8217;ll get:<\/p>\n<div class=\"examplesentence\">\\(s'(x)=-18x^{3}-9\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow, the bottom of the derivative equals \u201cthe square of what\u2019s below,\u201d so that\u2019ll be \\((x^{3}-1)^{2}\\).<\/p>\n<p>The derivative of \\(s(x)\\) is then:<\/p>\n<div class=\"examplesentence\">\\(s'(x)=\\frac{-18x^{3}-9}{(x^{3}-1)^{2}}\\)<\/div>\n<p>\n&nbsp;<\/p>\n<h3><span id=\"Example_2\" class=\"m-toc-anchor\"><\/span>Example 2<\/h3>\n<p>\nLet\u2019s try another example with the quotient rule.<\/p>\n<p>Find the derivative of \\(t(x)=\\frac{e^{x}}{sin(x)}\\). To find \\(t'(x)\\), we know we are going to have a fraction. The top will equal \u201cLow d High minus High d Low,\u201d so that\u2019s \\(sin(x)\\) times the derivative of \\(e^x\\), which is still \\(e^x\\), minus \\(e^{x}\\), times the derivative of \\(sin(x)\\), which is \\(cos(x)\\).<\/p>\n<div class=\"examplesentence\">\\(t'(x)=\\frac{sin(x)\\cdot e^x-e^x\\cdot cos(x)}{}\\)<\/div>\n<p>\n&nbsp;<br \/>\nThen, the bottom of the fraction will equal \u201clow squared,\u201d so \\(sin(x)\\) squared. <\/p>\n<div class=\"examplesentence\">\\(t'(x)=\\frac{sin(x)\\cdot e^x-e^x\\cdot cos(x)}{sin(x)^2}\\)<\/div>\n<p>\n&nbsp;<\/p>\n<p>Now remember, you can write \\(sin(x)^{2}\\) as \\(sin^{2}(x)\\); this is just a different way of writing the same thing.<\/p>\n<div class=\"examplesentence\">\\(t'(x)=\\frac{sin(x)\\cdot e^{x}-e^{x}\\cdot cos(x)}{sin^{2}(x)}\\)<\/div>\n<p>\n&nbsp;<br \/>\nWe can factor the numerator to clean it up a little, so that we have:<\/p>\n<div class=\"examplesentence\">\\(t'(x)=\\frac{e^{x}(sin(x)-cos(x))}{sin^{2}(x)}\\)<\/div>\n<p>\n&nbsp;<br \/>\nAnd that\u2019s our derivative!<\/p>\n<h3><span id=\"Example_3\" class=\"m-toc-anchor\"><\/span>Example 3<\/h3>\n<p>\nLet\u2019s try one last example. This one will require use of both the product rule and the quotient rule. <\/p>\n<p>Find the derivative of:<\/p>\n<div class=\"examplesentence\">\\(u(x)=\\frac{x\\text{ }sin(x)}{7x^{2}+2}\\)<\/div>\n<p>\n&nbsp;<br \/>\nNotice that this function is in fraction form, with a product in the numerator. Let\u2019s label the top \\(f\\) and the bottom \\(g\\) to make it easier to keep track of everything.<br \/>\n&nbsp;<\/p>\n<div class=\"examplesentence\">\\(f=x\\text{ }sin(x)\\)<br \/>\n&nbsp;<br \/>\n\\(g=7x^{2}+2\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow, the quotient rule tells us that the derivative\u2019s numerator equals \u201cLow d High minus High d Low,\u201d so let\u2019s go ahead and figure out what \\(f&#8217;\\) and \\(g&#8217;\\) are equal to now.<\/p>\n<p>To get \\(f&#8217;\\), we need to use the product rule. First, we write \\(x\\) times the derivative of \\(sin(x)\\), which is \\(cos(x)\\), and then we\u2019ll add \\(sin(x)\\) times the derivative of \\(x\\), which is 1.<\/p>\n<div class=\"examplesentence\">\\(f'((x)=x\\text{ }cos(x)+sin(x)\\)<\/div>\n<p>\n&nbsp;<br \/>\nSince \\(g\\) is a polynomial, \\(g&#8217;=14x\\):<\/p>\n<div class=\"examplesentence\">\\(g&#8217;=14x\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow that we have \\(f&#8217;\\) and \\(g&#8217;\\), we can use the quotient rule. \u201cLow d High\u201d means \\(gf&#8217;\\), so we start by writing:<\/p>\n<div class=\"examplesentence\">\\(u'(x)=(7x^{2}+2)(x\\text{ }cos(x)\\)\\(+sin(x))&#8230;\\)<\/div>\n<p>\n&nbsp;<br \/>\nThen, subtract \u201cHigh d Low\u201d or \\(fg&#8217;\\).<\/p>\n<div class=\"examplesentence\">\\(u'(x)=(7x^{2}+2)(x\\text{ }cos(x)\\)\\(+sin(x))-x\\text{ }sin(x)(14x)&#8230;\\)<\/div>\n<p>\n&nbsp;<br \/>\nThen the denominator of the derivative will be \\(g^{2}\\), which is \\((7x^2+2)^2\\)<\/p>\n<div class=\"examplesentence\">\\(u'(x)=\\)\\(\\frac{(7x^{2}+2)(x\\text{ }cos(x)+sin(x))-x\\text{ }sin(x)(14x)}{(7x^{2}+2)^2}\\)<\/div>\n<p>\n&nbsp;<br \/>\nThe product rule and quotient rule are fairly easy to work with once they are committed to memory, and the easiest way to memorize them is to work through some examples on your own.<\/p>\n<p>Thanks for watching, and happy studying!<\/p>\n<\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/calculus\/\">Return to Calculus Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Calculus Videos<\/p>\n","protected":false},"author":1,"featured_media":229393,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-37829","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-calculus-videos","7":"page_category-math-advertising-group","8":"page_category-video-pages-for-study-course-sidebar-ad","9":"page_type-video","10":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/37829","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=37829"}],"version-history":[{"count":5,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/37829\/revisions"}],"predecessor-version":[{"id":279436,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/37829\/revisions\/279436"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/229393"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=37829"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}