{"id":37,"date":"2021-02-03T20:17:58","date_gmt":"2021-02-03T20:17:58","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/the-derivative-as-a-function\/"},"modified":"2021-03-26T21:44:32","modified_gmt":"2021-03-26T21:44:32","slug":"the-derivative-as-a-function","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/the-derivative-as-a-function\/","title":{"raw":"Summary of the Derivative as a Function","rendered":"Summary of the Derivative as a Function"},"content":{"raw":"<div id=\"fs-id1169737935217\" class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1169737935224\">\r\n \t<li>The derivative of a function [latex]f(x)[\/latex] is the function whose value at [latex]x[\/latex] is [latex]f^{\\prime}(x)[\/latex].<\/li>\r\n \t<li>The graph of a derivative of a function [latex]f(x)[\/latex] is related to the graph of [latex]f(x)[\/latex]. Where [latex]f(x)[\/latex] has a tangent line with positive slope, [latex]f^{\\prime}(x)&gt;0[\/latex]. Where [latex]f(x)[\/latex] has a tangent line with negative slope, [latex]f^{\\prime}(x)&lt;0[\/latex]. Where [latex]f(x)[\/latex] has a horizontal tangent line, [latex]f^{\\prime}(x)=0[\/latex].<\/li>\r\n \t<li>If a function is differentiable at a point, then it is continuous at that point. A function is not differentiable at a point if it is not continuous at the point, if it has a vertical tangent line at the point, or if the graph has a sharp corner or cusp.<\/li>\r\n \t<li>Higher-order derivatives are derivatives of derivatives, from the second derivative to the [latex]n\\text{th}[\/latex] derivative.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1169738226133\" class=\"key-equations\">\r\n<h2>Key Equations<\/h2>\r\n<ul>\r\n \t<li><strong>The derivative function<\/strong>\r\n[latex]f^{\\prime}(x)=\\underset{h\\to 0}{\\lim}\\dfrac{f(x+h)-f(x)}{h}[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1169738071113\" class=\"definition\">\r\n \t<dt>derivative function<\/dt>\r\n \t<dd id=\"fs-id1169738071118\">gives the derivative of a function at each point in the domain of the original function for which the derivative is defined<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169738071124\" class=\"definition\">\r\n \t<dt>differentiable at [latex]a[\/latex]<\/dt>\r\n \t<dd id=\"fs-id1169738071129\">a function for which [latex]f^{\\prime}(a)[\/latex] exists is differentiable at [latex]a[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169738071160\" class=\"definition\">\r\n \t<dt>differentiable on [latex]S[\/latex]<\/dt>\r\n \t<dd id=\"fs-id1169738071165\">a function for which [latex]f^{\\prime}(x)[\/latex] exists for each [latex]x[\/latex] in the open set [latex]S[\/latex] is differentiable on [latex]S[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169738071204\" class=\"definition\">\r\n \t<dt>differentiable function<\/dt>\r\n \t<dd id=\"fs-id1169738071210\">a function for which [latex]f^{\\prime}(x)[\/latex] exists is a differentiable function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169738071234\" class=\"definition\">\r\n \t<dt>higher-order derivative<\/dt>\r\n \t<dd id=\"fs-id1169738071240\">a derivative of a derivative, from the second derivative to the [latex]n[\/latex]th derivative, is called a higher-order derivative<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<div id=\"fs-id1169737935217\" class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1169737935224\">\n<li>The derivative of a function [latex]f(x)[\/latex] is the function whose value at [latex]x[\/latex] is [latex]f^{\\prime}(x)[\/latex].<\/li>\n<li>The graph of a derivative of a function [latex]f(x)[\/latex] is related to the graph of [latex]f(x)[\/latex]. Where [latex]f(x)[\/latex] has a tangent line with positive slope, [latex]f^{\\prime}(x)>0[\/latex]. Where [latex]f(x)[\/latex] has a tangent line with negative slope, [latex]f^{\\prime}(x)<0[\/latex]. Where [latex]f(x)[\/latex] has a horizontal tangent line, [latex]f^{\\prime}(x)=0[\/latex].<\/li>\n<li>If a function is differentiable at a point, then it is continuous at that point. A function is not differentiable at a point if it is not continuous at the point, if it has a vertical tangent line at the point, or if the graph has a sharp corner or cusp.<\/li>\n<li>Higher-order derivatives are derivatives of derivatives, from the second derivative to the [latex]n\\text{th}[\/latex] derivative.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1169738226133\" class=\"key-equations\">\n<h2>Key Equations<\/h2>\n<ul>\n<li><strong>The derivative function<\/strong><br \/>\n[latex]f^{\\prime}(x)=\\underset{h\\to 0}{\\lim}\\dfrac{f(x+h)-f(x)}{h}[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1169738071113\" class=\"definition\">\n<dt>derivative function<\/dt>\n<dd id=\"fs-id1169738071118\">gives the derivative of a function at each point in the domain of the original function for which the derivative is defined<\/dd>\n<\/dl>\n<dl id=\"fs-id1169738071124\" class=\"definition\">\n<dt>differentiable at [latex]a[\/latex]<\/dt>\n<dd id=\"fs-id1169738071129\">a function for which [latex]f^{\\prime}(a)[\/latex] exists is differentiable at [latex]a[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1169738071160\" class=\"definition\">\n<dt>differentiable on [latex]S[\/latex]<\/dt>\n<dd id=\"fs-id1169738071165\">a function for which [latex]f^{\\prime}(x)[\/latex] exists for each [latex]x[\/latex] in the open set [latex]S[\/latex] is differentiable on [latex]S[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1169738071204\" class=\"definition\">\n<dt>differentiable function<\/dt>\n<dd id=\"fs-id1169738071210\">a function for which [latex]f^{\\prime}(x)[\/latex] exists is a differentiable function<\/dd>\n<\/dl>\n<dl id=\"fs-id1169738071234\" class=\"definition\">\n<dt>higher-order derivative<\/dt>\n<dd id=\"fs-id1169738071240\">a derivative of a derivative, from the second derivative to the [latex]n[\/latex]th derivative, is called a higher-order derivative<\/dd>\n<\/dl>\n<\/div>\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-37\">\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 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