{"id":41,"date":"2021-02-03T20:17:59","date_gmt":"2021-02-03T20:17:59","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/the-chain-rule\/"},"modified":"2021-04-02T22:55:27","modified_gmt":"2021-04-02T22:55:27","slug":"the-chain-rule","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/the-chain-rule\/","title":{"raw":"Summary of the Chain Rule","rendered":"Summary of the Chain Rule"},"content":{"raw":"<div id=\"fs-id1169736594101\" class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1169736594108\">\r\n \t<li>The chain rule allows us to differentiate compositions of two or more functions. It states that for [latex]h(x)=f(g(x))[\/latex],\r\n<div id=\"fs-id1169736594153\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]h^{\\prime}(x)=f^{\\prime}(g(x))g^{\\prime}(x)[\/latex]<\/div>\r\n<ul>\r\n \t<li>\r\n<div id=\"fs-id1169736594153\" class=\"equation unnumbered\" style=\"text-align: left;\">In Leibniz\u2019s notation this rule takes the form<\/div>\r\n<div id=\"fs-id1169737159954\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{dy}{dx}=\\frac{dy}{du} \\cdot \\frac{du}{dx}[\/latex]<\/div><\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>We can use the chain rule with other rules that we have learned, and we can derive formulas for some of them.<\/li>\r\n \t<li>The chain rule combines with the power rule to form a new rule:\r\n<div id=\"fs-id1169737470447\" class=\"equation unnumbered\" style=\"text-align: center;\">If [latex]h(x)=(g(x))^n[\/latex], then [latex]h^{\\prime}(x)=n(g(x))^{n-1}g^{\\prime}(x)[\/latex]<\/div><\/li>\r\n \t<li style=\"text-align: left;\">When applied to the composition of three functions, the chain rule can be expressed as follows: If [latex]h(x)=f(g(k(x)))[\/latex], then [latex]h^{\\prime}(x)=f^{\\prime}(g(k(x)))g^{\\prime}(k(x))k^{\\prime}(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1169736654969\" class=\"key-equations\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>The chain rule<\/strong>\r\n[latex]\\frac{d}{dx}(f(g(x)))=f^{\\prime}(g(x))g^{\\prime}(x)[\/latex]<\/li>\r\n \t<li><strong>The power rule for functions<\/strong>\r\n[latex]\\frac{d}{dx}((g(x)^n)=n(g(x))^{n-1}g^{\\prime}(x)[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1169739296745\" class=\"definition\">\r\n \t<dt>chain rule<\/dt>\r\n \t<dd id=\"fs-id1169739296750\">the chain rule defines the derivative of a composite function as the derivative of the outer function evaluated at the inner function times the derivative of the inner function<\/dd>\r\n<\/dl>","rendered":"<div id=\"fs-id1169736594101\" class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1169736594108\">\n<li>The chain rule allows us to differentiate compositions of two or more functions. It states that for [latex]h(x)=f(g(x))[\/latex],\n<div id=\"fs-id1169736594153\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]h^{\\prime}(x)=f^{\\prime}(g(x))g^{\\prime}(x)[\/latex]<\/div>\n<ul>\n<li>\n<div id=\"fs-id1169736594153\" class=\"equation unnumbered\" style=\"text-align: left;\">In Leibniz\u2019s notation this rule takes the form<\/div>\n<div id=\"fs-id1169737159954\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{dy}{dx}=\\frac{dy}{du} \\cdot \\frac{du}{dx}[\/latex]<\/div>\n<\/li>\n<\/ul>\n<\/li>\n<li>We can use the chain rule with other rules that we have learned, and we can derive formulas for some of them.<\/li>\n<li>The chain rule combines with the power rule to form a new rule:\n<div id=\"fs-id1169737470447\" class=\"equation unnumbered\" style=\"text-align: center;\">If [latex]h(x)=(g(x))^n[\/latex], then [latex]h^{\\prime}(x)=n(g(x))^{n-1}g^{\\prime}(x)[\/latex]<\/div>\n<\/li>\n<li style=\"text-align: left;\">When applied to the composition of three functions, the chain rule can be expressed as follows: If [latex]h(x)=f(g(k(x)))[\/latex], then [latex]h^{\\prime}(x)=f^{\\prime}(g(k(x)))g^{\\prime}(k(x))k^{\\prime}(x)[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1169736654969\" class=\"key-equations\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong>The chain rule<\/strong><br \/>\n[latex]\\frac{d}{dx}(f(g(x)))=f^{\\prime}(g(x))g^{\\prime}(x)[\/latex]<\/li>\n<li><strong>The power rule for functions<\/strong><br \/>\n[latex]\\frac{d}{dx}((g(x)^n)=n(g(x))^{n-1}g^{\\prime}(x)[\/latex]<\/li>\n<\/ul>\n<\/div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1169739296745\" class=\"definition\">\n<dt>chain rule<\/dt>\n<dd id=\"fs-id1169739296750\">the chain rule defines the derivative of a composite function as the derivative of the outer function evaluated at the inner function times the derivative of the inner function<\/dd>\n<\/dl>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-41\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 1. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\">https:\/\/openstax.org\/details\/books\/calculus-volume-1<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":28,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/details\/books\/calculus-volume-1\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-1\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-41","chapter","type-chapter","status-publish","hentry"],"part":35,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/41","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":14,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/41\/revisions"}],"predecessor-version":[{"id":2512,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/41\/revisions\/2512"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/35"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/41\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=41"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=41"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=41"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=41"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}