{"id":36,"date":"2021-02-03T20:17:57","date_gmt":"2021-02-03T20:17:57","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/defining-the-derivative\/"},"modified":"2021-04-02T21:27:15","modified_gmt":"2021-04-02T21:27:15","slug":"defining-the-derivative","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/defining-the-derivative\/","title":{"raw":"Summary of Defining the Derivative","rendered":"Summary of Defining the Derivative"},"content":{"raw":"<div id=\"fs-id1169739274316\" class=\"key-equations\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1169736611561\">\r\n \t<li>The slope of the tangent line to a curve measures the instantaneous rate of change of a curve. We can calculate it by finding the limit of the difference quotient or the difference quotient with increment [latex]h[\/latex].<\/li>\r\n \t<li>The derivative of a function [latex]f(x)[\/latex] at a value [latex]a[\/latex] is found using either of the definitions for the slope of the tangent line.<\/li>\r\n \t<li>Velocity is the rate of change of position. As such, the velocity [latex]v(t)[\/latex] at time [latex]t[\/latex] is the derivative of the position [latex]s(t)[\/latex] at time [latex]t[\/latex].\r\n<ul>\r\n \t<li>Average velocity is given by\r\n<div id=\"fs-id1169739286403\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]v_{\\text{avg}}=\\dfrac{s(t)-s(a)}{t-a}[\/latex]<\/div><\/li>\r\n \t<li>\r\n<div id=\"fs-id1169739286403\" class=\"equation unnumbered\" style=\"text-align: left;\">Instantaneous velocity is given by<\/div>\r\n<div id=\"fs-id1169739191149\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]v(a)=s^{\\prime}(a)=\\underset{t\\to a}{\\lim}\\dfrac{s(t)-s(a)}{t-a}[\/latex]<\/div><\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>We may estimate a derivative by using a table of values.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169739274323\">\r\n \t<li><strong>Difference quotient<\/strong>\r\n[latex]Q=\\dfrac{f(x)-f(a)}{x-a}[\/latex]\r\n<div><\/div><\/li>\r\n \t<li><strong>Difference quotient with increment\u00a0[latex]h[\/latex]<\/strong>\r\n[latex]Q=\\dfrac{f(a+h)-f(a)}{a+h-a}=\\dfrac{f(a+h)-f(a)}{h}[\/latex]\r\n<div><\/div><\/li>\r\n \t<li><strong>Slope of tangent line<\/strong>\r\n[latex]m_{\\tan}=\\underset{x\\to a}{\\lim}\\dfrac{f(x)-f(a)}{x-a}[\/latex]\r\n<div><\/div>\r\n[latex]m_{\\tan}=\\underset{h\\to 0}{\\lim}\\dfrac{f(a+h)-f(a)}{h}[\/latex]\r\n<div><\/div><\/li>\r\n \t<li><strong>Derivative of [latex]f(x)[\/latex] at [latex]a[\/latex]<\/strong>\r\n[latex]f^{\\prime}(a)=\\underset{x\\to a}{\\lim}\\dfrac{f(x)-f(a)}{x-a}[\/latex]\r\n<div><\/div>\r\n[latex]f^{\\prime}(a)=\\underset{h\\to 0}{\\lim}\\dfrac{f(a+h)-f(a)}{h}[\/latex]\r\n<div><\/div><\/li>\r\n \t<li><strong>Average velocity<\/strong>\r\n[latex]v_{\\text{avg}}=\\dfrac{s(t)-s(a)}{t-a}[\/latex]\r\n<div><\/div><\/li>\r\n \t<li><strong>Instantaneous velocity<\/strong>\r\n[latex]v(a)=s^{\\prime}(a)=\\underset{t\\to a}{\\lim}\\dfrac{s(t)-s(a)}{t-a}[\/latex]\r\n<div><\/div><\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1169739111084\" class=\"definition\">\r\n \t<dt>derivative<\/dt>\r\n \t<dd id=\"fs-id1169739111089\">the slope of the tangent line to a function at a point, calculated by taking the limit of the difference quotient, is the derivative<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169739111095\" class=\"definition\">\r\n \t<dt>difference quotient<\/dt>\r\n \t<dd id=\"fs-id1169739111100\">of a function [latex]f(x)[\/latex] at [latex]a[\/latex] is given by\r\n<div id=\"fs-id1169739111122\" class=\"equation unnumbered\">\r\n\r\n[latex]\\dfrac{f(a+h)-f(a)}{h}[\/latex] or [latex]\\dfrac{f(x)-f(a)}{x-a}[\/latex]\r\n<div><\/div>\r\n<\/div><\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169739111206\" class=\"definition\">\r\n \t<dt>differentiation<\/dt>\r\n \t<dd id=\"fs-id1169739111211\">the process of taking a derivative<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169739111216\" class=\"definition\">\r\n \t<dt>instantaneous rate of change<\/dt>\r\n \t<dd id=\"fs-id1169736619746\">the rate of change of a function at any point along the function [latex]a[\/latex], also called [latex]f^{\\prime}(a)[\/latex], or the derivative of the function at [latex]a[\/latex]<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<div id=\"fs-id1169739274316\" class=\"key-equations\">\n<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1169736611561\">\n<li>The slope of the tangent line to a curve measures the instantaneous rate of change of a curve. We can calculate it by finding the limit of the difference quotient or the difference quotient with increment [latex]h[\/latex].<\/li>\n<li>The derivative of a function [latex]f(x)[\/latex] at a value [latex]a[\/latex] is found using either of the definitions for the slope of the tangent line.<\/li>\n<li>Velocity is the rate of change of position. As such, the velocity [latex]v(t)[\/latex] at time [latex]t[\/latex] is the derivative of the position [latex]s(t)[\/latex] at time [latex]t[\/latex].\n<ul>\n<li>Average velocity is given by\n<div id=\"fs-id1169739286403\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]v_{\\text{avg}}=\\dfrac{s(t)-s(a)}{t-a}[\/latex]<\/div>\n<\/li>\n<li>\n<div id=\"fs-id1169739286403\" class=\"equation unnumbered\" style=\"text-align: left;\">Instantaneous velocity is given by<\/div>\n<div id=\"fs-id1169739191149\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]v(a)=s^{\\prime}(a)=\\underset{t\\to a}{\\lim}\\dfrac{s(t)-s(a)}{t-a}[\/latex]<\/div>\n<\/li>\n<\/ul>\n<\/li>\n<li>We may estimate a derivative by using a table of values.<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169739274323\">\n<li><strong>Difference quotient<\/strong><br \/>\n[latex]Q=\\dfrac{f(x)-f(a)}{x-a}[\/latex]<\/p>\n<div><\/div>\n<\/li>\n<li><strong>Difference quotient with increment\u00a0[latex]h[\/latex]<\/strong><br \/>\n[latex]Q=\\dfrac{f(a+h)-f(a)}{a+h-a}=\\dfrac{f(a+h)-f(a)}{h}[\/latex]<\/p>\n<div><\/div>\n<\/li>\n<li><strong>Slope of tangent line<\/strong><br \/>\n[latex]m_{\\tan}=\\underset{x\\to a}{\\lim}\\dfrac{f(x)-f(a)}{x-a}[\/latex]<\/p>\n<div><\/div>\n<p>[latex]m_{\\tan}=\\underset{h\\to 0}{\\lim}\\dfrac{f(a+h)-f(a)}{h}[\/latex]<\/p>\n<div><\/div>\n<\/li>\n<li><strong>Derivative of [latex]f(x)[\/latex] at [latex]a[\/latex]<\/strong><br \/>\n[latex]f^{\\prime}(a)=\\underset{x\\to a}{\\lim}\\dfrac{f(x)-f(a)}{x-a}[\/latex]<\/p>\n<div><\/div>\n<p>[latex]f^{\\prime}(a)=\\underset{h\\to 0}{\\lim}\\dfrac{f(a+h)-f(a)}{h}[\/latex]<\/p>\n<div><\/div>\n<\/li>\n<li><strong>Average velocity<\/strong><br \/>\n[latex]v_{\\text{avg}}=\\dfrac{s(t)-s(a)}{t-a}[\/latex]<\/p>\n<div><\/div>\n<\/li>\n<li><strong>Instantaneous velocity<\/strong><br \/>\n[latex]v(a)=s^{\\prime}(a)=\\underset{t\\to a}{\\lim}\\dfrac{s(t)-s(a)}{t-a}[\/latex]<\/p>\n<div><\/div>\n<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1169739111084\" class=\"definition\">\n<dt>derivative<\/dt>\n<dd id=\"fs-id1169739111089\">the slope of the tangent line to a function at a point, calculated by taking the limit of the difference quotient, is the derivative<\/dd>\n<\/dl>\n<dl id=\"fs-id1169739111095\" class=\"definition\">\n<dt>difference quotient<\/dt>\n<dd id=\"fs-id1169739111100\">of a function [latex]f(x)[\/latex] at [latex]a[\/latex] is given by<\/p>\n<div id=\"fs-id1169739111122\" class=\"equation unnumbered\">\n<p>[latex]\\dfrac{f(a+h)-f(a)}{h}[\/latex] or [latex]\\dfrac{f(x)-f(a)}{x-a}[\/latex]<\/p>\n<div><\/div>\n<\/div>\n<\/dd>\n<\/dl>\n<dl id=\"fs-id1169739111206\" class=\"definition\">\n<dt>differentiation<\/dt>\n<dd id=\"fs-id1169739111211\">the process of taking a derivative<\/dd>\n<\/dl>\n<dl id=\"fs-id1169739111216\" class=\"definition\">\n<dt>instantaneous rate of change<\/dt>\n<dd id=\"fs-id1169736619746\">the rate of change of a function at any point along the function [latex]a[\/latex], also called [latex]f^{\\prime}(a)[\/latex], or the derivative of the function at [latex]a[\/latex]<\/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-36\">\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":5,"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-36","chapter","type-chapter","status-publish","hentry"],"part":35,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/36","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":17,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/36\/revisions"}],"predecessor-version":[{"id":2450,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/36\/revisions\/2450"}],"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\/36\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=36"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=36"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=36"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=36"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}