Derivative of an Inverse. 1980 AB Free Response 3. Continuity and Differentiability of Inverses. If f is continuous in its domain, then its inverse is continuous on its domain. If f is increasing on its domain, then its inverse is increasing on its domain
Let’s investigate this…
If f is differentiable at c, the inverse is differentiable at f(c).
f is differentiable at x = 2.
Since f (2) = 6, g(x) is differentiable at x = 6.
If f '(c) = 0, the inverse is not differentiable at f(c).
f '(0) = 0
Since f (0) = 2, g(x) is not differentiable at x = 2.
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:
A function f and its derivative take on the values shown in the table.
If g is the inverse of f, find g'(6).
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)