{"id":1959,"date":"2021-03-24T23:37:40","date_gmt":"2021-03-24T23:37:40","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1959"},"modified":"2021-04-02T21:35:49","modified_gmt":"2021-04-02T21:35:49","slug":"summary-of-derivatives","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/summary-of-derivatives\/","title":{"raw":"Summary of Derivatives of Trigonometric Functions","rendered":"Summary of Derivatives of Trigonometric Functions"},"content":{"raw":"<div id=\"fs-id1169739325596\" class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1169739325602\">\r\n \t<li>We can find the derivatives of [latex]\\sin x[\/latex] and [latex]\\cos x[\/latex] by using the definition of derivative and the limit formulas found earlier. The results are\r\n<div id=\"fs-id1169736589261\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx} \\sin x= \\cos x[\/latex]\u00a0 and\u00a0 [latex]\\frac{d}{dx} \\cos x=\u2212\\sin x[\/latex].<\/div><\/li>\r\n \t<li>With these two formulas, we can determine the derivatives of all six basic trigonometric functions.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1169736603422\" class=\"key-equations\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169736603429\">\r\n \t<li><strong>Derivative of sine function<\/strong>\r\n[latex]\\frac{d}{dx}(\\sin x)= \\cos x[\/latex]<\/li>\r\n \t<li><strong>Derivative of cosine function<\/strong>\r\n[latex]\\frac{d}{dx}(\\cos x)=\u2212\\sin x[\/latex]<\/li>\r\n \t<li><strong>Derivative of tangent function<\/strong>\r\n[latex]\\frac{d}{dx}(\\tan x)=\\sec^2 x[\/latex]<\/li>\r\n \t<li><strong>Derivative of cotangent function<\/strong>\r\n[latex]\\frac{d}{dx}(\\cot x)=\u2212\\csc^2 x[\/latex]<\/li>\r\n \t<li><strong>Derivative of secant function<\/strong>\r\n[latex]\\frac{d}{dx}(\\sec x)= \\sec x \\tan x[\/latex]<\/li>\r\n \t<li><strong>Derivative of cosecant function<\/strong>\r\n[latex]\\frac{d}{dx}(\\csc x)=\u2212\\csc x \\cot x[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>","rendered":"<div id=\"fs-id1169739325596\" class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1169739325602\">\n<li>We can find the derivatives of [latex]\\sin x[\/latex] and [latex]\\cos x[\/latex] by using the definition of derivative and the limit formulas found earlier. The results are\n<div id=\"fs-id1169736589261\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\frac{d}{dx} \\sin x= \\cos x[\/latex]\u00a0 and\u00a0 [latex]\\frac{d}{dx} \\cos x=\u2212\\sin x[\/latex].<\/div>\n<\/li>\n<li>With these two formulas, we can determine the derivatives of all six basic trigonometric functions.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1169736603422\" class=\"key-equations\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736603429\">\n<li><strong>Derivative of sine function<\/strong><br \/>\n[latex]\\frac{d}{dx}(\\sin x)= \\cos x[\/latex]<\/li>\n<li><strong>Derivative of cosine function<\/strong><br \/>\n[latex]\\frac{d}{dx}(\\cos x)=\u2212\\sin x[\/latex]<\/li>\n<li><strong>Derivative of tangent function<\/strong><br \/>\n[latex]\\frac{d}{dx}(\\tan x)=\\sec^2 x[\/latex]<\/li>\n<li><strong>Derivative of cotangent function<\/strong><br \/>\n[latex]\\frac{d}{dx}(\\cot x)=\u2212\\csc^2 x[\/latex]<\/li>\n<li><strong>Derivative of secant function<\/strong><br \/>\n[latex]\\frac{d}{dx}(\\sec x)= \\sec x \\tan x[\/latex]<\/li>\n<li><strong>Derivative of cosecant function<\/strong><br \/>\n[latex]\\frac{d}{dx}(\\csc x)=\u2212\\csc x \\cot x[\/latex]<\/li>\n<\/ul>\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-1959\">\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":23,"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-1959","chapter","type-chapter","status-publish","hentry"],"part":35,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1959","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\/1959\/revisions"}],"predecessor-version":[{"id":2459,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1959\/revisions\/2459"}],"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\/1959\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1959"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1959"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1959"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1959"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}