14.7.1: Table of Derivatives


This table of derivatives is from another open textbook in Libretexts and OpenStax.

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

Chain rule

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

or

8. $$\quad \dfrac{dv}{dt} = \dfrac{ds}{dt} \dfrac{dv}{ds}$$

we will see this later when we note that acceleration is $$a = \dfrac{d^2s}{dt^2}$$

$a = \dfrac{d^2s}{dt^2} = \dfrac{ds}{dt}\dfrac{d(\frac{ds}{dt})}{ds} = v \dfrac{dv}{ds}$

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

Less Common Derivatives

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

For other derivatives you could use the computational intelligence website Wolframalpha.