Formula Sheet - Derivatives 5e1y17

Derivative Formulas In the following, u and v are functions of x, and n, e, a, and k are constants. 1. f '( x) = lim h →0

f ( x + h) − f ( x ) h

The Definition of the Derivative.

2.

d (k ) = 0 dx

The derivative of a constant is zero.

3.

d du k ( u ( x) ) ) = k ( dx dx

The derivative of a constant times a function.

4.

d n du u ) = nu n −1 ( dx dx

The Power Rule (Variable raised to a constant).

5.

d du dv (u + v ) = + dx dx dx

The Sum Rule.

6.

d du dv (u − v ) = − dx dx dx

The Difference Rule.

7.

d ( uv ) = uv '+ vu ' dx

The Product Rule.

8.

d ⎛ u ⎞ vu '− uv ' ⎜ ⎟= dx ⎝ v ⎠ v2

The Quotient Rule.

9.

dy dy du = dx du dx

The Chain Rule.

10.

d ( f ( g ( x) ) = f ' ( g ( x) ) g ' ( x ) dx

Another Form of the Chain Rule.

11.

d du ( sin u ) = cos u dx dx

The Derivative of the Sine.

12.

d du ( cos u ) = − sin u dx dx

The Derivative of the Cosine.

13.

d du ( tan u ) = sec2u dx dx

The Derivative of the Tangent.

14.

d du ( cot u ) = −csc2u dx dx

The Derivative of the Cotangent.

15.

d du ( sec u ) = sec u tan u dx dx

The Derivative of the Secant.

16.

d du ( csc u ) = − csc u cot u dx dx

The Derivative of the Cosecant.

17.

du d 1 Sin −1u ) = ( dx 1 − u 2 dx

The Derivative of the Inverse Sine.

18.

d −1 du Cos −1u ) = ( dx 1 − u 2 dx

The Derivative of the Inverse Cosine.

19.

d 1 du Tan −1u ) = ( dx 1 + u 2 dx

The Derivative of the Inverse Tangent.

20.

d −1 du Cot −1u ) = ( dx 1 + u 2 dx

The Derivative of the Inverse Cotangent.

21.

1 du d Sec −1u ) = ( 2 dx u u − 1 dx

The Derivative of the Inverse Secant.

22.

−1 du d Csc −1u ) = ( dx u u 2 − 1 dx

The Derivative of the Inverse Cosecant.

23.

d 1 du ( ln u ) = dx u dx

The Derivative of the Natural Log.

24.

d 1 du ( log a u ) = dx u ln a dx

The Derivative of the log to base a.

25.

d u du e ) = eu ( dx dx

The Derivative of e raised to a variable.

26.

d u du a ) = a u ln a ( dx dx

The Derivative of a constant raised to a variable.