{"id":1129,"date":"2021-03-06T03:08:07","date_gmt":"2021-03-06T03:08:07","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1129"},"modified":"2021-04-07T23:36:34","modified_gmt":"2021-04-07T23:36:34","slug":"summary-of-derivatives-of-exponential-and-logarithmic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/summary-of-derivatives-of-exponential-and-logarithmic-functions\/","title":{"raw":"Summary of Derivatives of Exponential and Logarithmic Functions","rendered":"Summary of Derivatives of Exponential and Logarithmic Functions"},"content":{"raw":"<div id=\"fs-id1169738233608\" class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1169738233616\">\r\n \t<li>On the basis of the assumption that the exponential function [latex]y=b^x, \\, b&gt;0[\/latex] is continuous everywhere and differentiable at 0, this function is differentiable everywhere and there is a formula for its derivative.<\/li>\r\n \t<li>We can use a formula to find the derivative of [latex]y=\\ln x[\/latex], and the relationship [latex]\\log_b x=\\dfrac{\\ln x}{\\ln b}[\/latex] allows us to extend our differentiation formulas to include logarithms with arbitrary bases.<\/li>\r\n \t<li>Logarithmic differentiation allows us to differentiate functions of the form [latex]y=g(x)^{f(x)}[\/latex] or very complex functions by taking the natural logarithm of both sides and exploiting the properties of logarithms before differentiating.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1169738234614\" class=\"key-equations\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169738234621\">\r\n \t<li><strong>Inverse function theorem<\/strong>\r\n[latex](f^{-1})^{\\prime}(x)=\\dfrac{1}{f^{\\prime}(f^{-1}(x))}[\/latex] whenever [latex]f^{\\prime}(f^{-1}(x))\\ne 0[\/latex] and [latex]f(x)[\/latex] is differentiable.<\/li>\r\n \t<li><strong>Power rule with rational exponents<\/strong>\r\n[latex]\\frac{d}{dx}(x^{m\/n})=\\frac{m}{n}x^{(m\/n)-1}[\/latex].<\/li>\r\n \t<li><strong>Derivative of the natural exponential function<\/strong>\r\n[latex]\\frac{d}{dx}(e^{g(x)})=e^{g(x)} g^{\\prime}(x)[\/latex]<\/li>\r\n \t<li><strong>Derivative of the natural logarithmic function<\/strong>\r\n[latex]\\frac{d}{dx}(\\ln (g(x)))=\\dfrac{1}{g(x)} g^{\\prime}(x)[\/latex]<\/li>\r\n \t<li><strong>Derivative of the general exponential function<\/strong>\r\n[latex]\\frac{d}{dx}(b^{g(x)})=b^{g(x)} g^{\\prime}(x) \\ln b[\/latex]<\/li>\r\n \t<li><strong>Derivative of the general logarithmic function<\/strong>\r\n[latex]\\frac{d}{dx}(\\log_b (g(x)))=\\dfrac{g^{\\prime}(x)}{g(x) \\ln b}[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1169738074914\" class=\"definition\">\r\n \t<dt>logarithmic differentiation<\/dt>\r\n \t<dd id=\"fs-id1169738074919\">is a technique that allows us to differentiate a function by first taking the natural logarithm of both sides of an equation, applying properties of logarithms to simplify the equation, and differentiating implicitly<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<div id=\"fs-id1169738233608\" class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1169738233616\">\n<li>On the basis of the assumption that the exponential function [latex]y=b^x, \\, b>0[\/latex] is continuous everywhere and differentiable at 0, this function is differentiable everywhere and there is a formula for its derivative.<\/li>\n<li>We can use a formula to find the derivative of [latex]y=\\ln x[\/latex], and the relationship [latex]\\log_b x=\\dfrac{\\ln x}{\\ln b}[\/latex] allows us to extend our differentiation formulas to include logarithms with arbitrary bases.<\/li>\n<li>Logarithmic differentiation allows us to differentiate functions of the form [latex]y=g(x)^{f(x)}[\/latex] or very complex functions by taking the natural logarithm of both sides and exploiting the properties of logarithms before differentiating.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1169738234614\" class=\"key-equations\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169738234621\">\n<li><strong>Inverse function theorem<\/strong><br \/>\n[latex](f^{-1})^{\\prime}(x)=\\dfrac{1}{f^{\\prime}(f^{-1}(x))}[\/latex] whenever [latex]f^{\\prime}(f^{-1}(x))\\ne 0[\/latex] and [latex]f(x)[\/latex] is differentiable.<\/li>\n<li><strong>Power rule with rational exponents<\/strong><br \/>\n[latex]\\frac{d}{dx}(x^{m\/n})=\\frac{m}{n}x^{(m\/n)-1}[\/latex].<\/li>\n<li><strong>Derivative of the natural exponential function<\/strong><br \/>\n[latex]\\frac{d}{dx}(e^{g(x)})=e^{g(x)} g^{\\prime}(x)[\/latex]<\/li>\n<li><strong>Derivative of the natural logarithmic function<\/strong><br \/>\n[latex]\\frac{d}{dx}(\\ln (g(x)))=\\dfrac{1}{g(x)} g^{\\prime}(x)[\/latex]<\/li>\n<li><strong>Derivative of the general exponential function<\/strong><br \/>\n[latex]\\frac{d}{dx}(b^{g(x)})=b^{g(x)} g^{\\prime}(x) \\ln b[\/latex]<\/li>\n<li><strong>Derivative of the general logarithmic function<\/strong><br \/>\n[latex]\\frac{d}{dx}(\\log_b (g(x)))=\\dfrac{g^{\\prime}(x)}{g(x) \\ln b}[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1169738074914\" class=\"definition\">\n<dt>logarithmic differentiation<\/dt>\n<dd id=\"fs-id1169738074919\">is a technique that allows us to differentiate a function by first taking the natural logarithm of both sides of an equation, applying properties of logarithms to simplify the equation, and differentiating implicitly<\/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-1129\">\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":40,"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-1129","chapter","type-chapter","status-publish","hentry"],"part":35,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1129","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":5,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1129\/revisions"}],"predecessor-version":[{"id":2830,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1129\/revisions\/2830"}],"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\/1129\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1129"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1129"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1129"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1129"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}