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

Objective Find the derivative of an inverse function.

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).

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.

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

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

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 .

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.

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

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