6.3: Table of Derivatives

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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)

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