{"id":2208,"date":"2018-01-11T21:35:59","date_gmt":"2018-01-11T21:35:59","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/back-matter\/table-of-derivatives\/"},"modified":"2018-01-11T21:35:59","modified_gmt":"2018-01-11T21:35:59","slug":"table-of-derivatives","status":"publish","type":"back-matter","link":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/back-matter\/table-of-derivatives\/","title":{"raw":"Table of Derivatives","rendered":"Table of Derivatives"},"content":{"raw":"

General Formulas<\/h1>

1. [latex]\\frac{d}{dx}(c)=0[\/latex]<\/p>

2. [latex]\\frac{d}{dx}(f(x)+g(x))={f}^{\\prime }(x)+{g}^{\\prime }(x)[\/latex]<\/p>

3. [latex]\\frac{d}{dx}(f(x)g(x))={f}^{\\prime }(x)g(x)+f(x){g}^{\\prime }(x)[\/latex]<\/p>

4. [latex]\\frac{d}{dx}({x}^{n})=n{x}^{n-1},\\text{for real numbers}n[\/latex]<\/p>

5. [latex]\\frac{d}{dx}(cf(x))=c{f}^{\\prime }(x)[\/latex]<\/p>

6. [latex]\\frac{d}{dx}(f(x)-g(x))={f}^{\\prime }(x)-{g}^{\\prime }(x)[\/latex]<\/p>

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>

8. [latex]\\frac{d}{dx}\\left[f(g(x))\\right]={f}^{\\prime }(g(x))\u00b7{g}^{\\prime }(x)[\/latex]<\/p><\/div>

Trigonometric Functions<\/h1>

9. [latex]\\frac{d}{dx}( \\sin x)= \\cos x[\/latex]<\/p>

10. [latex]\\frac{d}{dx}( \\tan x)={ \\sec }^{2}x[\/latex]<\/p>

11. [latex]\\frac{d}{dx}( \\sec x)= \\sec x \\tan x[\/latex]<\/p>

12. [latex]\\frac{d}{dx}( \\cos x)=\\text{\u2212} \\sin x[\/latex]<\/p>

13. [latex]\\frac{d}{dx}( \\cot x)=\\text{\u2212}{ \\csc }^{2}x[\/latex]<\/p>

14. [latex]\\frac{d}{dx}( \\csc x)=\\text{\u2212csc}x \\cot x[\/latex]<\/p><\/div>

Inverse Trigonometric Functions<\/h1>

15. [latex]\\frac{d}{dx}({ \\sin }^{-1}x)=\\frac{1}{\\sqrt{1-{x}^{2}}}[\/latex]<\/p>

16. [latex]\\frac{d}{dx}({ \\tan }^{-1}x)=\\frac{1}{1+{x}^{2}}[\/latex]<\/p>

17. [latex]\\frac{d}{dx}({ \\sec }^{-1}x)=\\frac{1}{|x|\\sqrt{{x}^{2}-1}}[\/latex]<\/p>

18. [latex]\\frac{d}{dx}({ \\cos }^{-1}x)=-\\frac{1}{\\sqrt{1-{x}^{2}}}[\/latex]<\/p>

19. [latex]\\frac{d}{dx}({ \\cot }^{-1}x)=-\\frac{1}{1+{x}^{2}}[\/latex]<\/p>

20. [latex]\\frac{d}{dx}({ \\csc }^{-1}x)=-\\frac{1}{|x|\\sqrt{{x}^{2}-1}}[\/latex]<\/p><\/div>

Exponential and Logarithmic Functions<\/h1>

21. [latex]\\frac{d}{dx}({e}^{x})={e}^{x}[\/latex]<\/p>

22. [latex]\\frac{d}{dx}(\\text{ln}|x|)=\\frac{1}{x}[\/latex]<\/p>

23. [latex]\\frac{d}{dx}({b}^{x})={b}^{x}\\text{ln}b[\/latex]<\/p>

24. [latex]\\frac{d}{dx}({\\text{log}}_{b}x)=\\frac{1}{x\\text{ln}b}[\/latex]<\/p><\/div>

Hyperbolic Functions<\/h1>

25. [latex]\\frac{d}{dx}(\\text{sinh}x)=\\text{cosh}x[\/latex]<\/p>

26. [latex]\\frac{d}{dx}(\\text{tanh}x)={\\text{sech}}^{2}x[\/latex]<\/p>

27. [latex]\\frac{d}{dx}(\\text{sech}x)=\\text{\u2212sech}x\\text{tanh}x[\/latex]<\/p>

28. [latex]\\frac{d}{dx}(\\text{cosh}x)=\\text{sinh}x[\/latex]<\/p>

29. [latex]\\frac{d}{dx}(\\text{coth}x)=\\text{\u2212}{\\text{csch}}^{2}x[\/latex]<\/p>

30. [latex]\\frac{d}{dx}(\\text{csch}x)=\\text{\u2212csch}x\\text{coth}x[\/latex]<\/p><\/div>

Inverse Hyperbolic Functions<\/h1>

31. [latex]\\frac{d}{dx}({\\text{sinh}}^{-1}x)=\\frac{1}{\\sqrt{{x}^{2}+1}}[\/latex]<\/p>

32. [latex]\\frac{d}{dx}({\\text{tanh}}^{-1}x)=\\frac{1}{1-{x}^{2}}(|x|<1)[\/latex]<\/p>

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>

34. [latex]\\frac{d}{dx}({\\text{cosh}}^{-1}x)=\\frac{1}{\\sqrt{{x}^{2}-1}}\\phantom{\\rule{1em}{0ex}}(x>1)[\/latex]<\/p>

35. [latex]\\frac{d}{dx}({\\text{coth}}^{-1}x)=\\frac{1}{1-{x}^{2}}\\phantom{\\rule{1em}{0ex}}(|x|>1)[\/latex]<\/p>

36. [latex]\\frac{d}{dx}({\\text{csch}}^{-1}x)=-\\frac{1}{|x|\\sqrt{1+{x}^{2}}}(x\\ne 0)[\/latex]<\/p><\/div>","rendered":"

\n

General Formulas<\/h1>\n

1. [latex]\\frac{d}{dx}(c)=0[\/latex]<\/p>\n

2. [latex]\\frac{d}{dx}(f(x)+g(x))={f}^{\\prime }(x)+{g}^{\\prime }(x)[\/latex]<\/p>\n

3. [latex]\\frac{d}{dx}(f(x)g(x))={f}^{\\prime }(x)g(x)+f(x){g}^{\\prime }(x)[\/latex]<\/p>\n

4. [latex]\\frac{d}{dx}({x}^{n})=n{x}^{n-1},\\text{for real numbers}n[\/latex]<\/p>\n

5. [latex]\\frac{d}{dx}(cf(x))=c{f}^{\\prime }(x)[\/latex]<\/p>\n

6. [latex]\\frac{d}{dx}(f(x)-g(x))={f}^{\\prime }(x)-{g}^{\\prime }(x)[\/latex]<\/p>\n

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

8. [latex]\\frac{d}{dx}\\left[f(g(x))\\right]={f}^{\\prime }(g(x))\u00b7{g}^{\\prime }(x)[\/latex]<\/p>\n<\/div>\n

\n

Trigonometric Functions<\/h1>\n

9. [latex]\\frac{d}{dx}( \\sin x)= \\cos x[\/latex]<\/p>\n

10. [latex]\\frac{d}{dx}( \\tan x)={ \\sec }^{2}x[\/latex]<\/p>\n

11. [latex]\\frac{d}{dx}( \\sec x)= \\sec x \\tan x[\/latex]<\/p>\n

12. [latex]\\frac{d}{dx}( \\cos x)=\\text{\u2212} \\sin x[\/latex]<\/p>\n

13. [latex]\\frac{d}{dx}( \\cot x)=\\text{\u2212}{ \\csc }^{2}x[\/latex]<\/p>\n

14. [latex]\\frac{d}{dx}( \\csc x)=\\text{\u2212csc}x \\cot x[\/latex]<\/p>\n<\/div>\n

\n

Inverse Trigonometric Functions<\/h1>\n

15. [latex]\\frac{d}{dx}({ \\sin }^{-1}x)=\\frac{1}{\\sqrt{1-{x}^{2}}}[\/latex]<\/p>\n

16. [latex]\\frac{d}{dx}({ \\tan }^{-1}x)=\\frac{1}{1+{x}^{2}}[\/latex]<\/p>\n

17. [latex]\\frac{d}{dx}({ \\sec }^{-1}x)=\\frac{1}{|x|\\sqrt{{x}^{2}-1}}[\/latex]<\/p>\n

18. [latex]\\frac{d}{dx}({ \\cos }^{-1}x)=-\\frac{1}{\\sqrt{1-{x}^{2}}}[\/latex]<\/p>\n

19. [latex]\\frac{d}{dx}({ \\cot }^{-1}x)=-\\frac{1}{1+{x}^{2}}[\/latex]<\/p>\n

20. [latex]\\frac{d}{dx}({ \\csc }^{-1}x)=-\\frac{1}{|x|\\sqrt{{x}^{2}-1}}[\/latex]<\/p>\n<\/div>\n

\n

Exponential and Logarithmic Functions<\/h1>\n

21. [latex]\\frac{d}{dx}({e}^{x})={e}^{x}[\/latex]<\/p>\n

22. [latex]\\frac{d}{dx}(\\text{ln}|x|)=\\frac{1}{x}[\/latex]<\/p>\n

23. [latex]\\frac{d}{dx}({b}^{x})={b}^{x}\\text{ln}b[\/latex]<\/p>\n

24. [latex]\\frac{d}{dx}({\\text{log}}_{b}x)=\\frac{1}{x\\text{ln}b}[\/latex]<\/p>\n<\/div>\n

\n

Hyperbolic Functions<\/h1>\n

25. [latex]\\frac{d}{dx}(\\text{sinh}x)=\\text{cosh}x[\/latex]<\/p>\n

26. [latex]\\frac{d}{dx}(\\text{tanh}x)={\\text{sech}}^{2}x[\/latex]<\/p>\n

27. [latex]\\frac{d}{dx}(\\text{sech}x)=\\text{\u2212sech}x\\text{tanh}x[\/latex]<\/p>\n

28. [latex]\\frac{d}{dx}(\\text{cosh}x)=\\text{sinh}x[\/latex]<\/p>\n

29. [latex]\\frac{d}{dx}(\\text{coth}x)=\\text{\u2212}{\\text{csch}}^{2}x[\/latex]<\/p>\n

30. [latex]\\frac{d}{dx}(\\text{csch}x)=\\text{\u2212csch}x\\text{coth}x[\/latex]<\/p>\n<\/div>\n

\n

Inverse Hyperbolic Functions<\/h1>\n

31. [latex]\\frac{d}{dx}({\\text{sinh}}^{-1}x)=\\frac{1}{\\sqrt{{x}^{2}+1}}[\/latex]<\/p>\n

32. [latex]\\frac{d}{dx}({\\text{tanh}}^{-1}x)=\\frac{1}{1-{x}^{2}}(|x|<1)[\/latex]<\/p>\n

33. [latex]\\frac{d}{dx}({\\text{sech}}^{-1}x)=-\\frac{1}{x\\sqrt{1-{x}^{2}}}\\phantom{\\rule{1em}{0ex}}(0\n

34. [latex]\\frac{d}{dx}({\\text{cosh}}^{-1}x)=\\frac{1}{\\sqrt{{x}^{2}-1}}\\phantom{\\rule{1em}{0ex}}(x>1)[\/latex]<\/p>\n

35. [latex]\\frac{d}{dx}({\\text{coth}}^{-1}x)=\\frac{1}{1-{x}^{2}}\\phantom{\\rule{1em}{0ex}}(|x|>1)[\/latex]<\/p>\n

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":311,"menu_order":2,"template":"","meta":{"_candela_citation":"","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"back-matter-type":[],"contributor":[],"license":[],"class_list":["post-2208","back-matter","type-back-matter","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/back-matter\/2208","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/back-matter"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/types\/back-matter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/users\/311"}],"version-history":[{"count":0,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/back-matter\/2208\/revisions"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/back-matter\/2208\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/media?parent=2208"}],"wp:term":[{"taxonomy":"back-matter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/back-matter-type?post=2208"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/contributor?post=2208"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/license?post=2208"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}