{"id":9290,"date":"2017-03-14t10:31:17","date_gmt":"2017-03-14t17:31:17","guid":{"rendered":"\/\/www.catharsisit.com\/hs\/?p=9290"},"modified":"2017-03-12t10:32:11","modified_gmt":"2017-03-12t17:32:11","slug":"ap-calculus-review-chain-rule","status":"publish","type":"post","link":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-chain-rule\/","title":{"rendered":"ap calculus review: chain rule"},"content":{"rendered":"<p>of all the derivative rules it seems that the chain rule gets the worst press.  many students dread the rule, think that it&#8217;s too difficult, don&#8217;t fully understand where to apply it, and generally wish that it would go away.  <\/p>\n<p>are you in this group?  <\/p>\n<p>if so then i hope that by the end of this short article, you&#8217;ll gain a better appreciation for the chain rule and how it is used in derivative problems.<\/p>\n<h2>the chain rule<\/h2>\n<p>we use the chain rule to find the derivative of a <strong>composition<\/strong> of functions, that is a function of the form <em>f<\/em>(<em>g<\/em>(<em>x<\/em>)).  <\/p>\n<h3>what is a composition?<\/h3>\n<p>if <em>f<\/em>(<em>x<\/em>) and <em>g<\/em>(<em>x<\/em>) are two functions, then we call <em>f<\/em>(<em>g<\/em>(<em>x<\/em>)) the composition of <em>f<\/em> and <em>g<\/em>.  we might call <em>f<\/em> the &#8220;outside&#8221; function, and <em>g<\/em> the &#8220;inside&#8221; function.<\/p>\n<p>for example, if <em>f<\/em>(<em>x<\/em>) = sin(<em>x<\/em>), and <em>g<\/em>(<em>x<\/em>) = <em>x<\/em><sup>2<\/sup> + 1, then their composition is:<\/p>\n<p><em>f<\/em>(<em>g<\/em>(<em>x<\/em>)) = sin(<em>x<\/em><sup>2<\/sup> + 1).<\/p>\n<h3>decomposition<\/h3>\n<p>to use the chain rule properly, you need to know how to go the other direction.  in other words, you have to learn how to <strong>decompose<\/strong> a composite function into a pair of functions &mdash; the outside and inside functions.<\/p>\n<p>to make it clearer, we often use a different variable name, such as <em>u<\/em>, for the input variable of the outside function.  <\/p>\n<p>for example, if <em>y<\/em> = (4<em>x<\/em> &#8211; 2)<sup>3<\/sup>, then the most natural decomposition would be:<\/p>\n<p><em>(outside function, or <em>f<\/em>(<em>u<\/em>)):<\/em> &nbsp;&nbsp;<em>y<\/em>  = <em>u<\/em><sup>3<\/sup>, and <\/p>\n<p><em>(inside function, or <em>g<\/em>(<em>x<\/em>)):<\/em> &nbsp;&nbsp;<em>u<\/em> = 4<em>x<\/em> &#8211; 2.<\/p>\n<h3>statement of the chain rule<\/h3>\n<p>suppose <em>f<\/em> and <em>g<\/em> are differentiable functions.  then the composite function, <em>f<\/em>(<em>g<\/em>(<em>x<\/em>)) is also differentiable, and<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/chainrule.gif\" alt=\"statement of the chain rule\" width=\"206\" height=\"20\" class=\"aligncenter size-full wp-image-9293\" \/><\/p>\n<p>i think of this as a <strong>3-step process<\/strong>.<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/three-step-process.jpeg\" alt=\"three step process for chain rule\" width=\"361\" height=\"195\" class=\"aligncenter size-full wp-image-9298\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/three-step-process.jpeg 361w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/three-step-process-300x162.jpeg 300w\" sizes=\"(max-width: 361px) 100vw, 361px\" \/><\/p>\n<ol>\n<li>find the derivative of the outside.<\/li>\n<li>plug in the inside function.<\/li>\n<li>multiply by the derivative of the inside.<\/li>\n<\/ol>\n<p>equivalently, if <em>y<\/em> = <em>f<\/em>(<em>u<\/em>), and <em>u<\/em> = <em>g<\/em>(<em>x<\/em>), then<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/chainrule_leibniz.gif\" alt=\"chain rule in leibniz notation\" width=\"103\" height=\"38\" class=\"aligncenter size-full wp-image-9294\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/chainrule_leibniz.gif 103w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/chainrule_leibniz-30x11.gif 30w\" sizes=\"(max-width: 103px) 100vw, 103px\" \/><\/p>\n<p>here, the rule has been written in leibniz notation, and clearly expresses the fact that the derivative of a composition is really just the product of the derivatives of the individual functions.  just don&#8217;t forget that step of plugging back in <em>u<\/em> = <em>g<\/em>(<em>x<\/em>).<\/p>\n<h3>example<\/h3>\n<p>find the derivative of <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/chainrule_example1.gif\" alt=\"y = square root of x^2 + 1\" width=\"98\" height=\"21\" class=\"alignnone size-full wp-image-9295\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/chainrule_example1.gif 98w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/chainrule_example1-30x6.gif 30w\" sizes=\"(max-width: 98px) 100vw, 98px\" \/>.<\/p>\n<p>first decompose the function.  <\/p>\n<p><em>(outside function):<\/em> &nbsp;&nbsp;<em>f<\/em>(<em>u<\/em>)  = <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/square_root1.gif\" alt=\"square root of u\" width=\"25\" height=\"19\" class=\"alignnone size-full wp-image-9297\" \/>. <\/p>\n<p><em>(inside function):<\/em> &nbsp;&nbsp;<em>g<\/em>(<em>x<\/em>) = <em>x<\/em><sup>2<\/sup> + 1.<\/p>\n<p>following the three-step process,<\/p>\n<ol>\n<li>first find the derivative of the outside: <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/derivative_of_square_root.gif\" alt=\"derivative of the outside function\" width=\"191\" height=\"43\" class=\"alignnone size-full wp-image-9299\" \/><\/li>\n<li>next, plug in the inside function: <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/step2_example_chain_rule.gif\" alt=\"plugging in the inside function\" width=\"73\" height=\"42\" class=\"alignnone size-full wp-image-9300\" \/><\/li>\n<li>finally, multiply by the derivative of the inside: <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/step3_example_chain_rule.gif\" alt=\"step 3 of the chain rule example\" width=\"120\" height=\"42\" class=\"aligncenter size-full wp-image-9301\" \/><\/li>\n<\/ol>\n<p>now that we have found the derivative correctly, let&#8217;s simplify the final answer using algebra.<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/chain_rule_example_solution.gif\" alt=\"final simplified form of the example derivative\" width=\"358\" height=\"43\" class=\"aligncenter size-full wp-image-9302\" \/><\/p>\n<h2>conclusion<\/h2>\n<p>use the chain rule to find the derivative of a composition.  always keep in mind the <em>3-step process<\/em>:<\/p>\n<ol>\n<li>find the derivative of the outside.<\/li>\n<li>plug in the inside function.<\/li>\n<li>multiply by the derivative of the inside.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>of all the derivative rules it seems that the chain rule gets the worst press. many students dread the rule, think that it&#8217;s too difficult, don&#8217;t fully understand where to apply it, and generally wish that it would go away. are you in this group? if so then i hope that by the end of [&hellip;]<\/p>\n","protected":false},"author":223,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[240],"tags":[241],"ppma_author":[24932],"acf":[],"yoast_head":"<!-- this site is optimized with the yoast seo premium plugin v21.7 (yoast seo v21.7) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>ap calculus review: chain rule - magoosh blog | high school<\/title>\n<meta name=\"description\" content=\"do you dread the chain rule, think that it&#039;s too difficult, or don&#039;t fully understand where to apply it? click here for a quick review of the chain rule.\" \/>\n<meta name=\"robots\" content=\"index, follow, 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