{"id":612,"date":"2025-10-09T15:25:37","date_gmt":"2025-10-09T15:25:37","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=612"},"modified":"2025-10-15T14:51:13","modified_gmt":"2025-10-15T14:51:13","slug":"advanced-partial-derivatives-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/advanced-partial-derivatives-cheat-sheet\/","title":{"raw":"Advanced Partial Derivatives: Cheat Sheet","rendered":"Advanced Partial Derivatives: Cheat Sheet"},"content":{"raw":"<h2>Essential Concepts<\/h2>\r\n<strong>The Chain Rule<\/strong>\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<strong>Tangent Planes and Linear Approximations<\/strong>\r\n<ul>\r\n \t<li>The analog of a tangent line to a curve is a tangent plane to a surface for functions of two variables.<\/li>\r\n \t<li>Tangent planes can be used to approximate values of functions near known values.<\/li>\r\n \t<li>A function is differentiable at a point if it is \u201dsmooth\u201d at that point (i.e., no corners or discontinuities exist at that point).<\/li>\r\n \t<li>The total differential can be used to approximate the change in a function [latex]z=f(x_{0},y_{0})[\/latex]\u00a0at the point\u00a0[latex](x_{0},y_{0})[\/latex]\u00a0for given values of [latex]\\Delta{x}[\/latex] and\u00a0[latex]\\Delta{y}[\/latex].<\/li>\r\n<\/ul>\r\n<strong>Directional Derivatives and the Gradient<\/strong>\r\n<ul id=\"fs-id1167793299690\" data-bullet-style=\"bullet\">\r\n \t<li>A directional derivative represents a rate of change of a function in any given direction.<\/li>\r\n \t<li>The gradient can be used in a formula to calculate the directional derivative.<\/li>\r\n \t<li>The gradient indicates the direction of greatest change of a function of more than one variable.<\/li>\r\n<\/ul>\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 \t<li><strong>Tangent plane\r\n<\/strong>[latex]z=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Linear approximation<\/strong>\r\n[latex]L(x,y)=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Total differential<\/strong>\r\n[latex]dz=f_{x}(x_{0},y_{0})dx+f_{y}(x_{0},y_{0})(y-y_{0})dy[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Differentiability (two variables)<\/strong>\r\n[latex]f(x,y)=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})+E(x,y)[\/latex], where the error term [latex]E[\/latex] satisfies [latex]\\underset{(x,y)\\to (x_{0},y_{0})}{\\lim}\\frac{E(x,y)}{\\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}}}=0[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Differentiability (three variables)<\/strong>\r\n[latex]f(x,y,z)=f(x_{0},y_{0},z_{0})+f_{x}(x_{0},y_{0},z_{0})(x-x_{0})+f_{y}(x_{0},y_{0},z_{0})(y-y_{0})+f_{z}(x_{0},y_{0},z_{0})(z-z_{0})+E(x,y,z)[\/latex], where the error term [latex]E[\/latex] satisfies [latex]\\underset{(x,y,z)\\to (x_{0},y_{0},z_{0})}{\\lim}\\frac{E(x,y,z)}{\\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}}}=0[\/latex]<\/li>\r\n \t<li><strong>Directional derivative (two dimensions)\r\n<\/strong>[latex]D_{\\bf{u}} f(a,b)=\\underset{h \\to {0}}{\\lim} \\frac{f(a+h\\cos{\\theta}, b+h\\sin{\\theta})-f(a,b)}{h}[\/latex]\u00a0 or\u00a0 [latex]D_{\\bf{u}} f(x,y)=f_{x}(x,y)\\cos{\\theta}+f_{y}(x,y)\\sin{\\theta}[\/latex]<\/li>\r\n \t<li><strong>Gradient (two dimensions)\r\n<\/strong>[latex]\\nabla f(x,y)=f_{x}(x,y){\\bf{i}}+f_{y}(x,y){\\bf{j}}[\/latex]<\/li>\r\n \t<li><strong>Gradient (three dimensions)\r\n<\/strong>[latex]\\nabla f(x,y,z)=f_{x}(x,y,z){\\bf{i}}+f_{y}(x,y,z){\\bf{j}}+f_{z}(x,y,z){\\bf{k}}[\/latex]<\/li>\r\n \t<li><strong>Directional derivative (three dimensions)\r\n<\/strong>[latex]\\begin{array}{c} D_{\\bf{u}} f(x,y,z) &amp;=\\nabla f(x,y,z)\\cdot{\\bf{u}} \\hfill \\\\ \\hfill &amp;=f_{x}(x,y,z)\\cos{\\alpha}+f_{y}(x,y,z)\\cos{\\beta}+f_{z}(x,y,z)\\cos{\\gamma}\\end{array}[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>differentiable<\/dt>\r\n \t<dd>a function [latex]f(x,y,z)[\/latex]\u00a0is differentiable at\u00a0[latex](x_{0},y_{0})[\/latex]\u00a0if [latex]f(x,y)[\/latex]\u00a0can be expressed in the form [latex]f(x,y)=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})+E(x,y)[\/latex], where the error term [latex]E(x,y)[\/latex]\u00a0satisfies [latex]\\underset{(x,y)\\to{(x_{0},y_{0})}}{\\lim}\\frac{E(x,y)}{\\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}}}=0[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>directional derivative<\/dt>\r\n \t<dd>the derivative of a function in the direction of a given unit vector<\/dd>\r\n<\/dl>\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>gradient<\/dt>\r\n \t<dd>the gradient of the function [latex]f(x,y)[\/latex]\u00a0is defined to be [latex]\\nabla f(x,y)=(\\partial{f}{\/}\\partial{x}){\\bf{i}}+(\\partial{f}{\/}\\partial{y}){\\bf{j}}[\/latex]\u00a0which can be generalized to a function of any number of independent 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>linear approximation<\/dt>\r\n \t<dd>given a function [latex]f(x,y)[\/latex]\u00a0and a tangent plane to the function at a point [latex](x_{0},y_{0})[\/latex] we can approximate [latex]f(x,y)[\/latex]\u00a0for points near [latex](x_{0},y_{0})[\/latex] using the tangent plane formula<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>tangent plane<\/dt>\r\n \t<dd>given a function [latex]f(x,y)[\/latex]\u00a0that is differentiable at a point [latex](x_{0},y_{0})[\/latex] the equation of the tangent plane to the surface [latex]z=f(x,y)[\/latex]\u00a0is given by [latex]z=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>total differential<\/dt>\r\n \t<dd>the total differential of the function [latex]f(x,y)[\/latex]\u00a0at [latex](x_{0},y_{0})[\/latex]\u00a0is given by the formula [latex]dz=f_{x}(x_{0},y_{0})dx+f_{y}(x_{0},y_{0})dy[\/latex]<\/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":"<h2>Essential Concepts<\/h2>\n<p><strong>The Chain Rule<\/strong><\/p>\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<p><strong>Tangent Planes and Linear Approximations<\/strong><\/p>\n<ul>\n<li>The analog of a tangent line to a curve is a tangent plane to a surface for functions of two variables.<\/li>\n<li>Tangent planes can be used to approximate values of functions near known values.<\/li>\n<li>A function is differentiable at a point if it is \u201dsmooth\u201d at that point (i.e., no corners or discontinuities exist at that point).<\/li>\n<li>The total differential can be used to approximate the change in a function [latex]z=f(x_{0},y_{0})[\/latex]\u00a0at the point\u00a0[latex](x_{0},y_{0})[\/latex]\u00a0for given values of [latex]\\Delta{x}[\/latex] and\u00a0[latex]\\Delta{y}[\/latex].<\/li>\n<\/ul>\n<p><strong>Directional Derivatives and the Gradient<\/strong><\/p>\n<ul id=\"fs-id1167793299690\" data-bullet-style=\"bullet\">\n<li>A directional derivative represents a rate of change of a function in any given direction.<\/li>\n<li>The gradient can be used in a formula to calculate the directional derivative.<\/li>\n<li>The gradient indicates the direction of greatest change of a function of more than one variable.<\/li>\n<\/ul>\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<li><strong>Tangent plane<br \/>\n<\/strong>[latex]z=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Linear approximation<\/strong><br \/>\n[latex]L(x,y)=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Total differential<\/strong><br \/>\n[latex]dz=f_{x}(x_{0},y_{0})dx+f_{y}(x_{0},y_{0})(y-y_{0})dy[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Differentiability (two variables)<\/strong><br \/>\n[latex]f(x,y)=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})+E(x,y)[\/latex], where the error term [latex]E[\/latex] satisfies [latex]\\underset{(x,y)\\to (x_{0},y_{0})}{\\lim}\\frac{E(x,y)}{\\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}}}=0[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Differentiability (three variables)<\/strong><br \/>\n[latex]f(x,y,z)=f(x_{0},y_{0},z_{0})+f_{x}(x_{0},y_{0},z_{0})(x-x_{0})+f_{y}(x_{0},y_{0},z_{0})(y-y_{0})+f_{z}(x_{0},y_{0},z_{0})(z-z_{0})+E(x,y,z)[\/latex], where the error term [latex]E[\/latex] satisfies [latex]\\underset{(x,y,z)\\to (x_{0},y_{0},z_{0})}{\\lim}\\frac{E(x,y,z)}{\\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}}}=0[\/latex]<\/li>\n<li><strong>Directional derivative (two dimensions)<br \/>\n<\/strong>[latex]D_{\\bf{u}} f(a,b)=\\underset{h \\to {0}}{\\lim} \\frac{f(a+h\\cos{\\theta}, b+h\\sin{\\theta})-f(a,b)}{h}[\/latex]\u00a0 or\u00a0 [latex]D_{\\bf{u}} f(x,y)=f_{x}(x,y)\\cos{\\theta}+f_{y}(x,y)\\sin{\\theta}[\/latex]<\/li>\n<li><strong>Gradient (two dimensions)<br \/>\n<\/strong>[latex]\\nabla f(x,y)=f_{x}(x,y){\\bf{i}}+f_{y}(x,y){\\bf{j}}[\/latex]<\/li>\n<li><strong>Gradient (three dimensions)<br \/>\n<\/strong>[latex]\\nabla f(x,y,z)=f_{x}(x,y,z){\\bf{i}}+f_{y}(x,y,z){\\bf{j}}+f_{z}(x,y,z){\\bf{k}}[\/latex]<\/li>\n<li><strong>Directional derivative (three dimensions)<br \/>\n<\/strong>[latex]\\begin{array}{c} D_{\\bf{u}} f(x,y,z) &=\\nabla f(x,y,z)\\cdot{\\bf{u}} \\hfill \\\\ \\hfill &=f_{x}(x,y,z)\\cos{\\alpha}+f_{y}(x,y,z)\\cos{\\beta}+f_{z}(x,y,z)\\cos{\\gamma}\\end{array}[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>differentiable<\/dt>\n<dd>a function [latex]f(x,y,z)[\/latex]\u00a0is differentiable at\u00a0[latex](x_{0},y_{0})[\/latex]\u00a0if [latex]f(x,y)[\/latex]\u00a0can be expressed in the form [latex]f(x,y)=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})+E(x,y)[\/latex], where the error term [latex]E(x,y)[\/latex]\u00a0satisfies [latex]\\underset{(x,y)\\to{(x_{0},y_{0})}}{\\lim}\\frac{E(x,y)}{\\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}}}=0[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>directional derivative<\/dt>\n<dd>the derivative of a function in the direction of a given unit vector<\/dd>\n<\/dl>\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>gradient<\/dt>\n<dd>the gradient of the function [latex]f(x,y)[\/latex]\u00a0is defined to be [latex]\\nabla f(x,y)=(\\partial{f}{\/}\\partial{x}){\\bf{i}}+(\\partial{f}{\/}\\partial{y}){\\bf{j}}[\/latex]\u00a0which can be generalized to a function of any number of independent 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>linear approximation<\/dt>\n<dd>given a function [latex]f(x,y)[\/latex]\u00a0and a tangent plane to the function at a point [latex](x_{0},y_{0})[\/latex] we can approximate [latex]f(x,y)[\/latex]\u00a0for points near [latex](x_{0},y_{0})[\/latex] using the tangent plane formula<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>tangent plane<\/dt>\n<dd>given a function [latex]f(x,y)[\/latex]\u00a0that is differentiable at a point [latex](x_{0},y_{0})[\/latex] the equation of the tangent plane to the surface [latex]z=f(x,y)[\/latex]\u00a0is given by [latex]z=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0})[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>total differential<\/dt>\n<dd>the total differential of the function [latex]f(x,y)[\/latex]\u00a0at [latex](x_{0},y_{0})[\/latex]\u00a0is given by the formula [latex]dz=f_{x}(x_{0},y_{0})dx+f_{y}(x_{0},y_{0})dy[\/latex]<\/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","protected":false},"author":15,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-612","chapter","type-chapter","status-publish","hentry"],"part":428,"module-header":"cheat_sheet","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/612","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/612\/revisions"}],"predecessor-version":[{"id":1162,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/612\/revisions\/1162"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/428"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/612\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=612"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=612"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=612"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=612"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}