6.2: Table of Derivatives

General Formulas

1. $\quad \dfrac{d}{dx}\left(c\right)=0$

2. $\quad \dfrac{d}{dx}\left(f(x)+g(x)\right)=f′(x)+g′(x)$

3. $\quad \dfrac{d}{dx}\left(f(x)g(x)\right)=f′(x)g(x)+f(x)g′(x)$

4. $\quad \dfrac{d}{dx}\left(x^n\right)=nx^{n−1},\quad \text{for real numbers }n$

5. $\quad \dfrac{d}{dx}\left(cf(x)\right)=cf′(x)$

6. $\quad \dfrac{d}{dx}\left(f(x)−g(x)\right)=f′(x)−g′(x)$

7. $\quad \dfrac{d}{dx}\left(\dfrac{f(x)}{g(x)}\right)=\dfrac{g(x)f′(x)−f(x)g′(x)}{(g(x))^2}$

8. $\quad \dfrac{d}{dx}\left[f(g(x))\right]=f′(g(x))·g′(x)$

Trigonometric Functions

9. $\quad \dfrac{d}{dx}\left(\sin x\right)=\cos x$

10. $\quad \dfrac{d}{dx}\left(\tan x\right)=\sec^2x$

11. $\quad \dfrac{d}{dx}\left(\sec x\right)=\sec x\tan x$

12. $\quad \dfrac{d}{dx}\left(\cos x\right)=−\sin x$

13. $\quad \dfrac{d}{dx}\left(\cot x\right)=−\csc^2x$

14. $\quad \dfrac{d}{dx}\left(\csc x\right)=−\csc x\cot x$

Inverse Trigonometric Functions

15. $\quad \dfrac{d}{dx}\left(\arcsin x\right)=\dfrac{1}{\sqrt{1−x^2}}$

16. $\quad \dfrac{d}{dx}\left(\arctan x\right)=\dfrac{1}{1+x^2}$

17. $\quad \dfrac{d}{dx}\left(\text{arcsec}\, x\right)=\dfrac{1}{|x|\sqrt{x^2−1}}$

18. $\quad \dfrac{d}{dx}\left(\arccos x\right)=\dfrac{-1}{\sqrt{1−x^2}}$

19. $\quad \dfrac{d}{dx}\left(\text{arccot}\, x\right)=\dfrac{-1}{1+x^2}$

20. $\quad \dfrac{d}{dx}\left(\text{arccsc}\, x\right)=\dfrac{-1}{|x|\sqrt{x^2−1}}$

Exponential and Logarithmic Functions

21. $\quad \dfrac{d}{dx}\left(e^x\right)=e^x$

22. $\quad \dfrac{d}{dx}\left(\ln|x|\right)=\dfrac{1}{x}$

23. $\quad \dfrac{d}{dx}\left(b^x\right)=b^x\ln b$

24. $\quad \dfrac{d}{dx}\left(\log_bx\right)=\dfrac{1}{x\ln b}$

Hyperbolic Functions

25. $\quad \dfrac{d}{dx}\left(\sinh x\right)=\cosh x$

26. $\quad \dfrac{d}{dx}\left(\tanh x\right)=\text{sech}^2 \,x$

27. $\quad \dfrac{d}{dx}\left(\text{sech} x\right)=−\text{sech} \,x\tanh x$

28. $\quad \dfrac{d}{dx}\left(\cosh x\right)=\sinh x$

29. $\quad \dfrac{d}{dx}\left(\coth x\right)=−\text{csch}^2 \,x$

30. $\quad \dfrac{d}{dx}\left(\text{csch} \,x\right)=−\text{csch} x\coth x$

Inverse Hyperbolic Functions

31. $\quad \dfrac{d}{dx}\left(\text{arcsinh}\, x\right)=\dfrac{1}{\sqrt{x^2+1}}$

32. $\quad \dfrac{d}{dx}\left(\text{arctanh}\, x\right)=\dfrac{1}{1-x^2}\quad (|x|<1)$

33. $\quad \dfrac{d}{dx}\left(\text{arcsech} \,x\right)=\dfrac{-1}{x\sqrt{1-x^2}}\quad (0<x<1)$

34. $\quad \dfrac{d}{dx}\left(\text{arccosh}\, x\right)=\dfrac{1}{\sqrt{x^2-1}}\quad (x>1)$

35. $\quad \dfrac{d}{dx}\left(\text{arccoth}\, x\right)=\dfrac{1}{1-x^2}\quad (|x|>1)$

36. $\quad \dfrac{d}{dx}\left(\text{arccsch}\,x\right)=\dfrac{-1}{|x|\sqrt{1+x^2}}\quad (x≠0)$

Contributors

• Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

• Modified to change inverse trig notation by Paul Seeburger (Monroe Community College)