Derivative Table

Derivative Table

1. d (u ? v) = du ? dv

dx

dx dx

2. d (cu) = c du

dx

dx

3. d (uv) = u dv + v du

dx

dx dx

4. d (uvw) = uv dw + vw du + wu dv

dx

dx

dx

dx

5.

d dx

u v

=

v

du - u dx

v2

dv dx

6. (Chain rule) If y = f(u) is differentiable on u = g(x) and u = g(x) is differentiable on point x, then the composite function y = f(g(x)) is differentiable and dy = dy du dx du dx

7. (Chain rule) dy = dy du dw dx du dw dx

8. (Inverse function) If y = f(x) has a non-zero derivative at x and the inverse function x = f -1(y) is continuous at corresponding point y, then x = f -1(y) is differentiable and:

dx = 1 dy dy

dx x = f (t)

9. (Parametric equation) For the equation y = g(t) , f(t) and g(t) are differentiable

dy and f'(t) 0, then dy = dt .

dx dx dt

10. (Parametric equation)

d2y

=

dx dt

d2y dt 2

- d2x dt 2

dy dt

=

x' y''-x'' y'

dx 2

dx 3

(x')3

dt

Page 1 of 3

11. dc = 0 dx

12. d x n = nx n-1 dx

13. d x = 1

dx

2x

14.

d dx

1 x

=

-

1 x2

15.

d dx

1 xn

=

-

n x n+1

16. d n x = 1

dx

n n x n-1

17. d ex = ex dx

18. d a x = a x ln a dx

19. d x x = x x (1 + ln x) dx

20. d ln x = 1

dx

x

21.

d dx

log a

x

=

1 x ln a

22. d log x = 1 log e 0.4343

dx

x

x

23. d sin x = cos x dx

24. d cos x = - sin x dx

25. d tan x = sec2 x dx

26. d sec x = sec x tan x dx

27. d cot x = - csc2 x dx

28. d csc x = - csc x cot x dx

29. d sin -1 x = 1

dx

1- x2

30. d cos-1 x = - 1

dx

1- x2

31.

d dx

tan

- 1

x

=

1

1 +x

2

32. d sec-1 x = 1

dx

x x2 -1

33.

d cot -1 x = - 1

dx

1+ x2

34. d csc-1 x = - 1

dx

x x2 -1

35. d sinh x = cosh x dx

36. d cosh x = sinh x dx

37. d tanh x = sec h 2x dx

38. d coth x = - csc h 2x dx

39. d sec h x = - sec h x tanh x dx

40. d csc hx = - csc h x coth x dx

( ) 41. d sinh -1 x = d ln x + 1 + x2 = 1

dx

dx

1+ x2

( ) 42. d cosh -1 x = d ln x + x 2 - 1 = ? 1 , x > 1

dx

dx

x2 -1

43.

d dx

tanh -1

x

=

d dx

1 2

ln

1+ 1-

x x

=

1 1- x2

,

x 1

45. d sec h -1x = ? 1 , x < 1

dx

x 1- x2

46. d csc h -1x = ? 1

dx

x x2 +1

47. d ln(sinh x) = coth x , d ln(cosh x) = tanh x

dx

dx

Page 2 of 3

Higher Derivatives

1.

dn dx n

xm

=

m(m - 1)....(m - n

+ 1)x m-n

2.

( ) dn ( ) dxn

x=

-

1

n

-1

1

?

3

?

5

?

.... 2n

?

2n - 3

- n- 1

x 2

3.

dn dx n

1 x

= (-1)n

n! x n+1

4.

dn dx n

ex

= ex

5.

dn dx n

eax+b

=

a n eax+b

6.

dn dx n

ax

=

a x (ln a)n

7.

dn ln x = (-1)n-1 (n -1)!

dx n

xn

8.

dn dx n

loga

x

=

(- )1 n-1

(n (ln

- 1)! a )x n

9.

dn dx n

sin

x

=

sin x

+

n 2

10.

dn dx n

cos

x

=

cos x

+

n 2

11.

dn dx n

sinh

x

=

sinh x cosh x

, n is even ,

, n is odd

dn dx n

cosh

x

=

cosh x sinh x

, n is even , n is odd

12.

dn dx n

sin 2

x

=

-2n-1

sin 2x

+

n 2

13.

dn dx n

sin

mx

=

mn

sin mx

+

n 2

14.

dn dx n

cos mx

=

mn

cos mx

+

n 2

15.

y = tan -1 x,

dny dx n

= (n - 1)!cosn

y sin ny +

n 2

16.

y = cot -1 x,

dny dx n

= (-1)n (n - 1)!sin n

y sin ny

17.

( ) y = eax sin bx,

dny dx n

=

a2

+ b2

n 2

eax

sin

bx

+

n

tan -1

b

a

18.

( ) y = eax cos bx,

dny dx n

=

a2 + b2

n 2

e ax

cos bx

+

n

tan -1

b

a

19. (Leibnitz Theorem)

(uv)(n) = n C n u(n-i)v(i) , i=0 i where u (0) = u, v(0) = v, u (r) = d r u dx r

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