Table of Derivatives

General Formulas

1. \frac{d}{dx}(c)=0

2. \frac{d}{dx}(f(x)+g(x))={f}^{\prime }(x)+{g}^{\prime }(x)

3. \frac{d}{dx}(f(x)g(x))={f}^{\prime }(x)g(x)+f(x){g}^{\prime }(x)

4. \frac{d}{dx}({x}^{n})=n{x}^{n-1},\text{for real numbers}n

5. \frac{d}{dx}(cf(x))=c{f}^{\prime }(x)

6. \frac{d}{dx}(f(x)-g(x))={f}^{\prime }(x)-{g}^{\prime }(x)

7. \frac{d}{dx}(\frac{f(x)}{g(x)})=\frac{g(x){f}^{\prime }(x)-f(x){g}^{\prime }(x)}{{(g(x))}^{2}}

8. \frac{d}{dx}\left[f(g(x))\right]={f}^{\prime }(g(x))·{g}^{\prime }(x)

Trigonometric Functions

9. \frac{d}{dx}( \sin x)= \cos x

10. \frac{d}{dx}( \tan x)={ \sec }^{2}x

11. \frac{d}{dx}( \sec x)= \sec x \tan x

12. \frac{d}{dx}( \cos x)=\text{−} \sin x

13. \frac{d}{dx}( \cot x)=\text{−}{ \csc }^{2}x

14. \frac{d}{dx}( \csc x)=\text{−csc}x \cot x

Inverse Trigonometric Functions

15. \frac{d}{dx}({ \sin }^{-1}x)=\frac{1}{\sqrt{1-{x}^{2}}}

16. \frac{d}{dx}({ \tan }^{-1}x)=\frac{1}{1+{x}^{2}}

17. \frac{d}{dx}({ \sec }^{-1}x)=\frac{1}{|x|\sqrt{{x}^{2}-1}}

18. \frac{d}{dx}({ \cos }^{-1}x)=-\frac{1}{\sqrt{1-{x}^{2}}}

19. \frac{d}{dx}({ \cot }^{-1}x)=-\frac{1}{1+{x}^{2}}

20. \frac{d}{dx}({ \csc }^{-1}x)=-\frac{1}{|x|\sqrt{{x}^{2}-1}}

Exponential and Logarithmic Functions

21. \frac{d}{dx}({e}^{x})={e}^{x}

22. \frac{d}{dx}(\text{ln}|x|)=\frac{1}{x}

23. \frac{d}{dx}({b}^{x})={b}^{x}\text{ln}b

24. \frac{d}{dx}({\text{log}}_{b}x)=\frac{1}{x\text{ln}b}

Hyperbolic Functions

25. \frac{d}{dx}(\text{sinh}x)=\text{cosh}x

26. \frac{d}{dx}(\text{tanh}x)={\text{sech}}^{2}x

27. \frac{d}{dx}(\text{sech}x)=\text{−sech}x\text{tanh}x

28. \frac{d}{dx}(\text{cosh}x)=\text{sinh}x

29. \frac{d}{dx}(\text{coth}x)=\text{−}{\text{csch}}^{2}x

30. \frac{d}{dx}(\text{csch}x)=\text{−csch}x\text{coth}x

Inverse Hyperbolic Functions

31. \frac{d}{dx}({\text{sinh}}^{-1}x)=\frac{1}{\sqrt{{x}^{2}+1}}

32. \frac{d}{dx}({\text{tanh}}^{-1}x)=\frac{1}{1-{x}^{2}}(|x|<1)

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

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

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

36. \frac{d}{dx}({\text{csch}}^{-1}x)=-\frac{1}{|x|\sqrt{1+{x}^{2}}}(x\ne 0)

License

Icon for the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License

Table of Derivatives by OSCRiceUniversity is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

Share This Book