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Mathematics LibreTexts

8.3: Table of Derivatives

  1. CCBYNCSA
  2. Last updated
    May 12, 2023
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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)

8.3: Table of Derivatives is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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