{"id":168,"date":"2021-07-30T17:24:00","date_gmt":"2021-07-30T17:24:00","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=168"},"modified":"2022-10-29T02:04:58","modified_gmt":"2022-10-29T02:04:58","slug":"summary-of-the-chain-rule","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-the-chain-rule\/","title":{"raw":"Summary of the Chain Rule","rendered":"Summary of the Chain Rule"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1167793398389\" data-bullet-style=\"bullet\">\r\n \t<li>The chain rule for functions of more than one variable involves the partial derivatives with respect to all the independent variables.<\/li>\r\n \t<li>Tree diagrams are useful for deriving formulas for the chain rule for functions of more than one variable, where each independent variable also depends on other variables.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>Chain rule, one independent variable\r\n<\/strong>[latex]\\dfrac{dz}{dt}=\\dfrac{\\partial z}{\\partial x}\\cdot\\dfrac{dx}{dt}+\\dfrac{\\partial z}{\\partial y}\\cdot\\dfrac{dy}{dt}[\/latex]<\/li>\r\n \t<li><strong>Chain rule, two independent variables<\/strong>\r\n[latex]\\begin{array}{c} \\dfrac{\\partial z}{\\partial u} &amp;=\\dfrac{\\partial z}{\\partial x}\\dfrac{\\partial x}{\\partial u}+\\dfrac{\\partial z}{\\partial y}\\dfrac{\\partial y}{\\partial u} \\\\\\dfrac{\\partial z}{\\partial v} &amp;=\\dfrac{\\partial z}{\\partial x}\\dfrac{\\partial x}{\\partial v}+\\dfrac{\\partial z}{\\partial y}\\dfrac{\\partial y}{\\partial v} \\end{array}[\/latex]<\/li>\r\n \t<li><strong>Generalized chain rule<\/strong>\r\n[latex]\\dfrac{\\partial w}{\\partial t_{j}}=\\dfrac{\\partial w}{\\partial x_{1}}\\dfrac{\\partial x_{1}}{\\partial t_{j}}+\\dfrac{\\partial w}{\\partial x_{2}}\\dfrac{\\partial x_{1}}{\\partial t_{j}}+\\ldots+\\dfrac{\\partial w}{\\partial x_{m}}\\dfrac{\\partial x_{m}}{\\partial t_{j}}[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>generalized chain rule<\/dt>\r\n \t<dd>the chain rule extended to functions of more than one independent variable, in which each independent variable may depend on one or more other variables<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>intermediate variable<\/dt>\r\n \t<dd>given a composition of functions (e.g., [latex]f\\left(x(t),y(t)\\right)[\/latex]) the intermediate variables are the variables that are independent in the outer function but dependent on other variables as well; in the function\u00a0<span style=\"font-size: 1em;\">[latex]f\\left(x(t),y(t)\\right)[\/latex]<\/span>\u00a0the variables [latex]x[\/latex]\u00a0and\u00a0[latex]y[\/latex]<strong>\u00a0<\/strong>are examples of intermediate variables<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>tree diagram<\/dt>\r\n \t<dd>illustrates and derives formulas for the generalized chain rule, in which each independent variable is accounted for<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1167793398389\" data-bullet-style=\"bullet\">\n<li>The chain rule for functions of more than one variable involves the partial derivatives with respect to all the independent variables.<\/li>\n<li>Tree diagrams are useful for deriving formulas for the chain rule for functions of more than one variable, where each independent variable also depends on other variables.<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong>Chain rule, one independent variable<br \/>\n<\/strong>[latex]\\dfrac{dz}{dt}=\\dfrac{\\partial z}{\\partial x}\\cdot\\dfrac{dx}{dt}+\\dfrac{\\partial z}{\\partial y}\\cdot\\dfrac{dy}{dt}[\/latex]<\/li>\n<li><strong>Chain rule, two independent variables<\/strong><br \/>\n[latex]\\begin{array}{c} \\dfrac{\\partial z}{\\partial u} &=\\dfrac{\\partial z}{\\partial x}\\dfrac{\\partial x}{\\partial u}+\\dfrac{\\partial z}{\\partial y}\\dfrac{\\partial y}{\\partial u} \\\\\\dfrac{\\partial z}{\\partial v} &=\\dfrac{\\partial z}{\\partial x}\\dfrac{\\partial x}{\\partial v}+\\dfrac{\\partial z}{\\partial y}\\dfrac{\\partial y}{\\partial v} \\end{array}[\/latex]<\/li>\n<li><strong>Generalized chain rule<\/strong><br \/>\n[latex]\\dfrac{\\partial w}{\\partial t_{j}}=\\dfrac{\\partial w}{\\partial x_{1}}\\dfrac{\\partial x_{1}}{\\partial t_{j}}+\\dfrac{\\partial w}{\\partial x_{2}}\\dfrac{\\partial x_{1}}{\\partial t_{j}}+\\ldots+\\dfrac{\\partial w}{\\partial x_{m}}\\dfrac{\\partial x_{m}}{\\partial t_{j}}[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>generalized chain rule<\/dt>\n<dd>the chain rule extended to functions of more than one independent variable, in which each independent variable may depend on one or more other variables<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>intermediate variable<\/dt>\n<dd>given a composition of functions (e.g., [latex]f\\left(x(t),y(t)\\right)[\/latex]) the intermediate variables are the variables that are independent in the outer function but dependent on other variables as well; in the function\u00a0<span style=\"font-size: 1em;\">[latex]f\\left(x(t),y(t)\\right)[\/latex]<\/span>\u00a0the variables [latex]x[\/latex]\u00a0and\u00a0[latex]y[\/latex]<strong>\u00a0<\/strong>are examples of intermediate variables<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>tree diagram<\/dt>\n<dd>illustrates and derives formulas for the generalized chain rule, in which each independent variable is accounted for<\/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-168\">\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 3. <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\/books\/calculus-volume-3\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/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-3\/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":23,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/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-168","chapter","type-chapter","status-publish","hentry"],"part":22,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/168","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":12,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/168\/revisions"}],"predecessor-version":[{"id":3767,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/168\/revisions\/3767"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/22"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/168\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=168"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=168"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=168"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=168"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}