1. 5.3 Inverse Function (part 2)
Lewis and Clark Caverns, Montana
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 1993

2. Objective
Find the derivative of an inverse function.

3. Derivative of an Inverse Function
Theorem 5.8: Continuity and Differentiability of Inverse Functions
If f has an inverse:
If f is continuous on its domain, then f -1 is continuous on its domain.
If f is increasing on its domain, then f -1 is increasing on its domain.
If f is decreasing on its domain, then f -1 is decreasing on its domain.
If f is differentiable at c and f’(c)≠0, then f-1 is differentiable at f(c).

4. The Derivative of an Inverse Function
Theorem 5.9:
If f is differentiable on an interval I and has an inverse function g, then g is differentiable at any x for which f '(g(x))≠0.

5. At x = 2:
We can find the inverse function as follows:
To find the derivative of the inverse function:
Switch x and y.

6. Slopes are reciprocals.
At x = 2:
At x = 4:

7. Slopes are reciprocals.
Because x and y are reversed to find the reciprocal function, the following pattern always holds:
The derivative of
Derivative Formula for Inverses:
evaluated at
is equal to the reciprocal of
the derivative of
evaluated at

8. Example Let a) What is the value of f-1(x) when x=3? y-value in f
When x=2, f(x)=3  f-1(x)=2.
So when x=3, f-1(x)=2.

9. Example (continued)
When
b) Find (f-1)'(x) when x=3?

10. Homework
5.3 (page 349)
#71-81 odd, 85