Derivative of an Inverse

1. Derivative of an Inverse

2. 1980 AB Free Response 3

3. Continuity and Differentiability of Inverses • If fis continuous in its domain, then its inverse is continuous on its domain. • If fis increasing on its domain, then its inverse is increasing on its domain • If fis decreasing on its domain, then its inverse is decreasing on its domain • If fis differentiable on an interval containing cand f '(c) does NOT equal 0, then the inverse is differentiable at f (c). Let’s investigate this…

4. Differentiability of an Inverse If f is differentiable at c, the inverse is differentiable at f(c). f is differentiable at x = 2. Example: Since f (2) = 6, g(x) is differentiable at x = 6. Reciprocals. 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.

5. 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 xfor which f '(g(x)) ≠ 0. In particular: Other Forms:

6. Example 1 A function f and its derivative take on the values shown in the table. If g is the inverse of f, find g'(6).

7. 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). (Solve x3+ x – 2 = 0 with a calculator or guess and check)

8. 2007 AB Free Response 3