Derivative of an Inverse

1980 AB Free Response 3

Continuity and Differentiability of Inverses

1. If f is continuous in its domain, then its inverse is continuous on its domain.

2. If f is increasing on its domain, then its inverse is increasing on its domain

3. If f is decreasing on its domain, then its inverse is decreasing on its domain

4. If f is differentiable on an interval containing c and f '(c) does NOT equal 0, then the inverse is differentiable at f (c).

Let’s investigate this…

Differentiability of an Inverse 2 2f x x

2g x x

f is differentiable at x = 2. ' 2 4f Since f (2) = 6, g(x) is differentiable at x = 6.

14' 6g

If f is differentiable at c, the inverse is differentiable at f(c).

Example:

If f '(c) = 0, the inverse is not differentiable at f(c).

Example:f '(0) = 0

Since f (0) = 2, g(x) is not differentiable at x = 2.

Reciprocals.

The Derivative of an Inverse

Assume that f(x) is differentiable and one-to-one on an interval I with inverse g(x). g(x) is differentiable at any x for which f '(g(x)) ≠ 0. In particular:

1

''

g xf g x

Other Forms:

1

1

1'

'f x

f f x

1'

'g b

f a

If ,f a b

1, ' ,

'f x

g f x

Example 1A function f and its derivative take on the values shown in the table.

x f (x) f '(x)

2 6 1/3

6 8 3/2

If g is the inverse of f, find g'(6).

' 6g

1

' 6f g

1

' 2f

1

1 33

6 2 since 2 6g f

Example 2

Let f (x) = x3 + x – 2 and let g be the inverse function. Evaluate g'(0).

Note: It is difficult to find an equation for the inverse function g. We NEED the formula to evaluate g'(0).

' 0g 1

' 0f g

0 1 since 1 0g f (Solve x3 + x – 2 = 0 with a calculator or guess and check)

2' 3 1f x x

1' 1f

2

1

3 1 1 1

4

2007 AB Free Response 3