{"id":367,"date":"2024-10-01T18:45:49","date_gmt":"2024-10-01T18:45:49","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/appendix-b-table-of-derivatives\/"},"modified":"2025-12-17T15:18:22","modified_gmt":"2025-12-17T15:18:22","slug":"appendix-b-table-of-derivatives","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/appendix-b-table-of-derivatives\/","title":{"raw":"Appendix B: Table of Derivatives","rendered":"Appendix B: Table of Derivatives"},"content":{"raw":"\n<div id=\"fs-id1165042332308\" class=\"bc-section section\">\n<h2>General Formulas<\/h2>\n<p id=\"fs-id1165042323771\">1. [latex]\\frac{d}{dx}(c)=0[\/latex]<\/p>\n<p id=\"fs-id1165043097261\">2. [latex]\\frac{d}{dx}(f(x)+g(x))={f}^{\\prime }(x)+{g}^{\\prime }(x)[\/latex]<\/p>\n<p id=\"fs-id1165042407696\">3. [latex]\\frac{d}{dx}(f(x)g(x))={f}^{\\prime }(x)g(x)+f(x){g}^{\\prime }(x)[\/latex]<\/p>\n<p id=\"fs-id1165042709614\">4. [latex]\\frac{d}{dx}({x}^{n})=n{x}^{n-1},\\text{for real numbers}n[\/latex]<\/p>\n<p id=\"fs-id1165043187354\">5. [latex]\\frac{d}{dx}(cf(x))=c{f}^{\\prime }(x)[\/latex]<\/p>\n<p id=\"fs-id1165043065765\">6. [latex]\\frac{d}{dx}(f(x)-g(x))={f}^{\\prime }(x)-{g}^{\\prime }(x)[\/latex]<\/p>\n<p id=\"fs-id1165042319007\">7. [latex]\\frac{d}{dx}(\\frac{f(x)}{g(x)})=\\frac{g(x){f}^{\\prime }(x)-f(x){g}^{\\prime }(x)}{{(g(x))}^{2}}[\/latex]<\/p>\n<p id=\"fs-id1165043226174\">8. [latex]\\frac{d}{dx}\\left[f(g(x))\\right]={f}^{\\prime }(g(x))\u00b7{g}^{\\prime }(x)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165043021286\" class=\"bc-section section\">\n<h2>Trigonometric Functions<\/h2>\n<p id=\"fs-id1165043163938\">9. [latex]\\frac{d}{dx}( \\sin x)= \\cos x[\/latex]<\/p>\n<p id=\"fs-id1165043253938\">10. [latex]\\frac{d}{dx}( \\tan x)={ \\sec }^{2}x[\/latex]<\/p>\n<p id=\"fs-id1165043380012\">11. [latex]\\frac{d}{dx}( \\sec x)= \\sec x \\tan x[\/latex]<\/p>\n<p id=\"fs-id1165042328088\">12. [latex]\\frac{d}{dx}( \\cos x)=\\text{\u2212} \\sin x[\/latex]<\/p>\n<p id=\"fs-id1165043353625\">13. [latex]\\frac{d}{dx}( \\cot x)=\\text{\u2212}{ \\csc }^{2}x[\/latex]<\/p>\n14. [latex]\\frac{d}{dx}( \\csc x)=\\text{\u2212csc}x \\cot x[\/latex]\n\n<\/div>\n<div id=\"fs-id1165043006501\" class=\"bc-section section\">\n<h2>Inverse Trigonometric Functions<\/h2>\n<p id=\"fs-id1165043035592\">15. [latex]\\frac{d}{dx}({ \\sin }^{-1}x)=\\frac{1}{\\sqrt{1-{x}^{2}}}[\/latex]<\/p>\n<p id=\"fs-id1165043423645\">16. [latex]\\frac{d}{dx}({ \\tan }^{-1}x)=\\frac{1}{1+{x}^{2}}[\/latex]<\/p>\n<p id=\"fs-id1165042987130\">17. [latex]\\frac{d}{dx}({ \\sec }^{-1}x)=\\frac{1}{|x|\\sqrt{{x}^{2}-1}}[\/latex]<\/p>\n<p id=\"fs-id1165043166481\">18. [latex]\\frac{d}{dx}({ \\cos }^{-1}x)=-\\frac{1}{\\sqrt{1-{x}^{2}}}[\/latex]<\/p>\n<p id=\"fs-id1165042320730\">19. [latex]\\frac{d}{dx}({ \\cot }^{-1}x)=-\\frac{1}{1+{x}^{2}}[\/latex]<\/p>\n<p id=\"fs-id1165043033414\">20. [latex]\\frac{d}{dx}({ \\csc }^{-1}x)=-\\frac{1}{|x|\\sqrt{{x}^{2}-1}}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165042318824\" class=\"bc-section section\">\n<h2>Exponential and Logarithmic Functions<\/h2>\n<p id=\"fs-id1165043256945\">21. [latex]\\frac{d}{dx}({e}^{x})={e}^{x}[\/latex]<\/p>\n<p id=\"fs-id1165043010410\">22. [latex]\\frac{d}{dx}(\\text{ln}|x|)=\\frac{1}{x}[\/latex]<\/p>\n<p id=\"fs-id1165043431956\">23. [latex]\\frac{d}{dx}({b}^{x})={b}^{x}\\text{ln}b[\/latex]<\/p>\n<p id=\"fs-id1165043119122\">24. [latex]\\frac{d}{dx}({\\text{log}}_{b}x)=\\frac{1}{x\\text{ln}b}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165043066683\" class=\"bc-section section\">\n<h2>Hyperbolic Functions<\/h2>\n<p id=\"fs-id1165043423336\">25. [latex]\\frac{d}{dx}(\\text{sinh}x)=\\text{cosh}x[\/latex]<\/p>\n<p id=\"fs-id1165043109532\">26. [latex]\\frac{d}{dx}(\\text{tanh}x)={\\text{sech}}^{2}x[\/latex]<\/p>\n<p id=\"fs-id1165042429175\">27. [latex]\\frac{d}{dx}(\\text{sech}x)=\\text{\u2212sech}x\\text{tanh}x[\/latex]<\/p>\n<p id=\"fs-id1165042988637\">28. [latex]\\frac{d}{dx}(\\text{cosh}x)=\\text{sinh}x[\/latex]<\/p>\n<p id=\"fs-id1165043085741\">29. [latex]\\frac{d}{dx}(\\text{coth}x)=\\text{\u2212}{\\text{csch}}^{2}x[\/latex]<\/p>\n<p id=\"fs-id1165043021681\">30. [latex]\\frac{d}{dx}(\\text{csch}x)=\\text{\u2212csch}x\\text{coth}x[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165043380702\" class=\"bc-section section\">\n<h2>Inverse Hyperbolic Functions<\/h2>\n<p id=\"fs-id1165043184028\">31. [latex]\\frac{d}{dx}({\\text{sinh}}^{-1}x)=\\frac{1}{\\sqrt{{x}^{2}+1}}[\/latex]<\/p>\n<p id=\"fs-id1165043036397\">32. [latex]\\frac{d}{dx}({\\text{tanh}}^{-1}x)=\\frac{1}{1-{x}^{2}}(|x|&lt;1)[\/latex]<\/p>\n<p id=\"fs-id1165043380719\">33. [latex]\\frac{d}{dx}({\\text{sech}}^{-1}x)=-\\frac{1}{x\\sqrt{1-{x}^{2}}}\\phantom{\\rule{1em}{0ex}}(0&lt;x&lt;1)[\/latex]<\/p>\n<p id=\"fs-id1165042330766\">34. [latex]\\frac{d}{dx}({\\text{cosh}}^{-1}x)=\\frac{1}{\\sqrt{{x}^{2}-1}}\\phantom{\\rule{1em}{0ex}}(x&gt;1)[\/latex]<\/p>\n<p id=\"fs-id1165042965149\">35. [latex]\\frac{d}{dx}({\\text{coth}}^{-1}x)=\\frac{1}{1-{x}^{2}}\\phantom{\\rule{1em}{0ex}}(|x|&gt;1)[\/latex]<\/p>\n<p id=\"fs-id1165042333082\">36. [latex]\\frac{d}{dx}({\\text{csch}}^{-1}x)=-\\frac{1}{|x|\\sqrt{1+{x}^{2}}}(x\\ne 0)[\/latex]<\/p>\n\n<\/div>\n","rendered":"<div id=\"fs-id1165042332308\" class=\"bc-section section\">\n<h2>General Formulas<\/h2>\n<p id=\"fs-id1165042323771\">1. [latex]\\frac{d}{dx}(c)=0[\/latex]<\/p>\n<p id=\"fs-id1165043097261\">2. [latex]\\frac{d}{dx}(f(x)+g(x))={f}^{\\prime }(x)+{g}^{\\prime }(x)[\/latex]<\/p>\n<p id=\"fs-id1165042407696\">3. [latex]\\frac{d}{dx}(f(x)g(x))={f}^{\\prime }(x)g(x)+f(x){g}^{\\prime }(x)[\/latex]<\/p>\n<p id=\"fs-id1165042709614\">4. [latex]\\frac{d}{dx}({x}^{n})=n{x}^{n-1},\\text{for real numbers}n[\/latex]<\/p>\n<p id=\"fs-id1165043187354\">5. [latex]\\frac{d}{dx}(cf(x))=c{f}^{\\prime }(x)[\/latex]<\/p>\n<p id=\"fs-id1165043065765\">6. [latex]\\frac{d}{dx}(f(x)-g(x))={f}^{\\prime }(x)-{g}^{\\prime }(x)[\/latex]<\/p>\n<p id=\"fs-id1165042319007\">7. [latex]\\frac{d}{dx}(\\frac{f(x)}{g(x)})=\\frac{g(x){f}^{\\prime }(x)-f(x){g}^{\\prime }(x)}{{(g(x))}^{2}}[\/latex]<\/p>\n<p id=\"fs-id1165043226174\">8. [latex]\\frac{d}{dx}\\left[f(g(x))\\right]={f}^{\\prime }(g(x))\u00b7{g}^{\\prime }(x)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165043021286\" class=\"bc-section section\">\n<h2>Trigonometric Functions<\/h2>\n<p id=\"fs-id1165043163938\">9. [latex]\\frac{d}{dx}( \\sin x)= \\cos x[\/latex]<\/p>\n<p id=\"fs-id1165043253938\">10. [latex]\\frac{d}{dx}( \\tan x)={ \\sec }^{2}x[\/latex]<\/p>\n<p id=\"fs-id1165043380012\">11. [latex]\\frac{d}{dx}( \\sec x)= \\sec x \\tan x[\/latex]<\/p>\n<p id=\"fs-id1165042328088\">12. [latex]\\frac{d}{dx}( \\cos x)=\\text{\u2212} \\sin x[\/latex]<\/p>\n<p id=\"fs-id1165043353625\">13. [latex]\\frac{d}{dx}( \\cot x)=\\text{\u2212}{ \\csc }^{2}x[\/latex]<\/p>\n<p>14. [latex]\\frac{d}{dx}( \\csc x)=\\text{\u2212csc}x \\cot x[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165043006501\" class=\"bc-section section\">\n<h2>Inverse Trigonometric Functions<\/h2>\n<p id=\"fs-id1165043035592\">15. [latex]\\frac{d}{dx}({ \\sin }^{-1}x)=\\frac{1}{\\sqrt{1-{x}^{2}}}[\/latex]<\/p>\n<p id=\"fs-id1165043423645\">16. [latex]\\frac{d}{dx}({ \\tan }^{-1}x)=\\frac{1}{1+{x}^{2}}[\/latex]<\/p>\n<p id=\"fs-id1165042987130\">17. [latex]\\frac{d}{dx}({ \\sec }^{-1}x)=\\frac{1}{|x|\\sqrt{{x}^{2}-1}}[\/latex]<\/p>\n<p id=\"fs-id1165043166481\">18. [latex]\\frac{d}{dx}({ \\cos }^{-1}x)=-\\frac{1}{\\sqrt{1-{x}^{2}}}[\/latex]<\/p>\n<p id=\"fs-id1165042320730\">19. [latex]\\frac{d}{dx}({ \\cot }^{-1}x)=-\\frac{1}{1+{x}^{2}}[\/latex]<\/p>\n<p id=\"fs-id1165043033414\">20. [latex]\\frac{d}{dx}({ \\csc }^{-1}x)=-\\frac{1}{|x|\\sqrt{{x}^{2}-1}}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165042318824\" class=\"bc-section section\">\n<h2>Exponential and Logarithmic Functions<\/h2>\n<p id=\"fs-id1165043256945\">21. [latex]\\frac{d}{dx}({e}^{x})={e}^{x}[\/latex]<\/p>\n<p id=\"fs-id1165043010410\">22. [latex]\\frac{d}{dx}(\\text{ln}|x|)=\\frac{1}{x}[\/latex]<\/p>\n<p id=\"fs-id1165043431956\">23. [latex]\\frac{d}{dx}({b}^{x})={b}^{x}\\text{ln}b[\/latex]<\/p>\n<p id=\"fs-id1165043119122\">24. [latex]\\frac{d}{dx}({\\text{log}}_{b}x)=\\frac{1}{x\\text{ln}b}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165043066683\" class=\"bc-section section\">\n<h2>Hyperbolic Functions<\/h2>\n<p id=\"fs-id1165043423336\">25. [latex]\\frac{d}{dx}(\\text{sinh}x)=\\text{cosh}x[\/latex]<\/p>\n<p id=\"fs-id1165043109532\">26. [latex]\\frac{d}{dx}(\\text{tanh}x)={\\text{sech}}^{2}x[\/latex]<\/p>\n<p id=\"fs-id1165042429175\">27. [latex]\\frac{d}{dx}(\\text{sech}x)=\\text{\u2212sech}x\\text{tanh}x[\/latex]<\/p>\n<p id=\"fs-id1165042988637\">28. [latex]\\frac{d}{dx}(\\text{cosh}x)=\\text{sinh}x[\/latex]<\/p>\n<p id=\"fs-id1165043085741\">29. [latex]\\frac{d}{dx}(\\text{coth}x)=\\text{\u2212}{\\text{csch}}^{2}x[\/latex]<\/p>\n<p id=\"fs-id1165043021681\">30. [latex]\\frac{d}{dx}(\\text{csch}x)=\\text{\u2212csch}x\\text{coth}x[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165043380702\" class=\"bc-section section\">\n<h2>Inverse Hyperbolic Functions<\/h2>\n<p id=\"fs-id1165043184028\">31. [latex]\\frac{d}{dx}({\\text{sinh}}^{-1}x)=\\frac{1}{\\sqrt{{x}^{2}+1}}[\/latex]<\/p>\n<p id=\"fs-id1165043036397\">32. [latex]\\frac{d}{dx}({\\text{tanh}}^{-1}x)=\\frac{1}{1-{x}^{2}}(|x|<1)[\/latex]<\/p>\n<p id=\"fs-id1165043380719\">33. [latex]\\frac{d}{dx}({\\text{sech}}^{-1}x)=-\\frac{1}{x\\sqrt{1-{x}^{2}}}\\phantom{\\rule{1em}{0ex}}(0<x<1)[\/latex]<\/p>\n<p id=\"fs-id1165042330766\">34. [latex]\\frac{d}{dx}({\\text{cosh}}^{-1}x)=\\frac{1}{\\sqrt{{x}^{2}-1}}\\phantom{\\rule{1em}{0ex}}(x>1)[\/latex]<\/p>\n<p id=\"fs-id1165042965149\">35. [latex]\\frac{d}{dx}({\\text{coth}}^{-1}x)=\\frac{1}{1-{x}^{2}}\\phantom{\\rule{1em}{0ex}}(|x|>1)[\/latex]<\/p>\n<p id=\"fs-id1165042333082\">36. [latex]\\frac{d}{dx}({\\text{csch}}^{-1}x)=-\\frac{1}{|x|\\sqrt{1+{x}^{2}}}(x\\ne 0)[\/latex]<\/p>\n<\/div>\n","protected":false},"author":6,"menu_order":3,"template":"","meta":{"_candela_citation":"{\"1\":{\"type\":\"cc\",\"description\":\"Calculus Volume 2\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\"}}","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":22,"module-header":"","content_attributions":{"1":{"type":"cc","description":"Calculus Volume 2","author":"Gilbert Strang, Edwin (Jed) Herman","organization":"OpenStax","url":"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction","project":"","license":"cc-by-nc-sa","license_terms":"Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction"}},"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\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/367"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":1,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/367\/revisions"}],"predecessor-version":[{"id":2508,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/367\/revisions\/2508"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/22"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/367\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=367"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=367"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=367"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=367"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}