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General formulas

1. d d x ( c ) = 0

2. d d x ( f ( x ) + g ( x ) ) = f ( x ) + g ( x )

3. d d x ( f ( x ) g ( x ) ) = f ( x ) g ( x ) + f ( x ) g ( x )

4. d d x ( x n ) = n x n 1 , for real numbers n

5. d d x ( c f ( x ) ) = c f ( x )

6. d d x ( f ( x ) g ( x ) ) = f ( x ) g ( x )

7. d d x ( f ( x ) g ( x ) ) = g ( x ) f ( x ) f ( x ) g ( x ) ( g ( x ) ) 2

8. d d x [ f ( g ( x ) ) ] = f ( g ( x ) ) · g ( x )

Trigonometric functions

9. d d x ( sin x ) = cos x

10. d d x ( tan x ) = sec 2 x

11. d d x ( sec x ) = sec x tan x

12. d d x ( cos x ) = sin x

13. d d x ( cot x ) = csc 2 x

14. d d x ( csc x ) = −csc x cot x

Inverse trigonometric functions

15. d d x ( sin −1 x ) = 1 1 x 2

16. d d x ( tan −1 x ) = 1 1 + x 2

17. d d x ( sec −1 x ) = 1 | x | x 2 1

18. d d x ( cos −1 x ) = 1 1 x 2

19. d d x ( cot −1 x ) = 1 1 + x 2

20. d d x ( csc −1 x ) = 1 | x | x 2 1

Exponential and logarithmic functions

21. d d x ( e x ) = e x

22. d d x ( ln | x | ) = 1 x

23. d d x ( b x ) = b x ln b

24. d d x ( log b x ) = 1 x ln b

Hyperbolic functions

25. d d x ( sinh x ) = cosh x

26. d d x ( tanh x ) = sech 2 x

27. d d x ( sech x ) = −sech x tanh x

28. d d x ( cosh x ) = sinh x

29. d d x ( coth x ) = csch 2 x

30. d d x ( csch x ) = −csch x coth x

Inverse hyperbolic functions

31. d d x ( sinh −1 x ) = 1 x 2 + 1

32. d d x ( tanh −1 x ) = 1 1 x 2 ( | x | < 1 )

33. d d x ( sech −1 x ) = 1 x 1 x 2 ( 0 < x < 1 )

34. d d x ( cosh −1 x ) = 1 x 2 1 ( x > 1 )

35. d d x ( coth −1 x ) = 1 1 x 2 ( | x | > 1 )

36. d d x ( csch −1 x ) = 1 | x | 1 + x 2 ( x 0 )

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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