Differentiating the Inverse

1. Differentiating the Inverse

2. Objectives Students will be able to • Calculate the inverse of a function. • Determine if a function has an inverse. • Differentiate the inverse of a given function. • Find the equation of the tangent line to the inverse of a given function at a given point.

3. Example 1 Find the inverse of the function Algebraically verify that the function f(x) and the function f-1(x) are inverses of each other.

4. Example 2-1 Determine if the function f(x) is one-to-one, continuous, and strictly increase or strictly decreasing.

5. Example 2-2 Find the inverse g(x) for the function f(x) below

6. Example 2-3 Find the derivative of f(x) (below) and g(x) (the inverse of f (x)).

7. Inverse Function Theorem If f is continuous and strictly increasing (or strictly decreasing) in an interval I, then f has an inverse function g, which is continuous and strictly increasing (strictly decreasing) in the interval f(I). If x0 is an interior point of I and f‘ (x0)≠0, then g is differentiable at y0 = f(x0) and

8. Example 2-4 Verify that the Inverse Function Theorem applies to the functions f(x) and g(x) at x0 = 2

9. Example 2-5 Find the equation of the tangent line to the curve g(x) at point (7, 2)