{"id":164,"date":"2021-07-30T17:23:28","date_gmt":"2021-07-30T17:23:28","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=164"},"modified":"2022-10-29T01:50:55","modified_gmt":"2022-10-29T01:50:55","slug":"summary-of-partial-derivatives","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-partial-derivatives\/","title":{"raw":"Summary of Partial Derivatives","rendered":"Summary of Partial Derivatives"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<div class=\"os-section-area\"><section id=\"fs-id1167793432260\" class=\"key-concepts\" data-depth=\"1\">\r\n<ul id=\"fs-id1167794160198\" data-bullet-style=\"bullet\">\r\n \t<li>A partial derivative is a derivative involving a function of more than one independent variable.<\/li>\r\n \t<li>To calculate a partial derivative with respect to a given variable, treat all the other variables as constants and use the usual differentiation rules.<\/li>\r\n \t<li>Higher-order partial derivatives can be calculated in the same way as higher-order derivatives.<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul>\r\n \t<li><strong>Partial derivative of [latex]f[\/latex]\u00a0with respect to\u00a0[latex]x[\/latex]\r\n<\/strong>[latex]\\frac{\\partial f}{\\partial x}=\\underset{h \\to{0}}{\\lim} \\frac{f(x+h,y)-f(x,y)}{h}[\/latex]<\/li>\r\n \t<li><strong>Partial derivative of [latex]f[\/latex]\u00a0with respect to\u00a0[latex]y[\/latex]\r\n<\/strong>[latex]\\frac{\\partial f}{\\partial y}=\\underset{k \\to{0}}{\\lim} \\frac{f(x,y+k)-f(x,y)}{k}[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>higher-order partial derivatives<\/dt>\r\n \t<dd>second-order or higher partial derivatives, regardless of whether they are mixed partial derivatives<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>mixed partial derivatives<\/dt>\r\n \t<dd>second-order or higher partial derivatives, in which at least two of the differentiations are with respect to different variables<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>partial derivative<\/dt>\r\n \t<dd>a derivative of a function of more than one independent variable in which all the variables but one are held constant<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>partial differential equation<\/dt>\r\n \t<dd>an equation that involves an unknown function of more than one independent variable and one or more of its partial derivatives<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<div class=\"os-section-area\">\n<section id=\"fs-id1167793432260\" class=\"key-concepts\" data-depth=\"1\">\n<ul id=\"fs-id1167794160198\" data-bullet-style=\"bullet\">\n<li>A partial derivative is a derivative involving a function of more than one independent variable.<\/li>\n<li>To calculate a partial derivative with respect to a given variable, treat all the other variables as constants and use the usual differentiation rules.<\/li>\n<li>Higher-order partial derivatives can be calculated in the same way as higher-order derivatives.<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul>\n<li><strong>Partial derivative of [latex]f[\/latex]\u00a0with respect to\u00a0[latex]x[\/latex]<br \/>\n<\/strong>[latex]\\frac{\\partial f}{\\partial x}=\\underset{h \\to{0}}{\\lim} \\frac{f(x+h,y)-f(x,y)}{h}[\/latex]<\/li>\n<li><strong>Partial derivative of [latex]f[\/latex]\u00a0with respect to\u00a0[latex]y[\/latex]<br \/>\n<\/strong>[latex]\\frac{\\partial f}{\\partial y}=\\underset{k \\to{0}}{\\lim} \\frac{f(x,y+k)-f(x,y)}{k}[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>higher-order partial derivatives<\/dt>\n<dd>second-order or higher partial derivatives, regardless of whether they are mixed partial derivatives<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>mixed partial derivatives<\/dt>\n<dd>second-order or higher partial derivatives, in which at least two of the differentiations are with respect to different variables<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>partial derivative<\/dt>\n<dd>a derivative of a function of more than one independent variable in which all the variables but one are held constant<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>partial differential equation<\/dt>\n<dd>an equation that involves an unknown function of more than one independent variable and one or more of its partial derivatives<\/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-164\">\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":15,"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-164","chapter","type-chapter","status-publish","hentry"],"part":22,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/164","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":9,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/164\/revisions"}],"predecessor-version":[{"id":3765,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/164\/revisions\/3765"}],"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\/164\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=164"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=164"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=164"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=164"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}