Objectives: In this tutorial, we investigate the differentiablility of an inverse function. We demonstrate that a function that has a derivative which is either positive for all x in an interval or negative for all x in an interval has an inverse that is also differentiable. We obtain a formula for the derivative of the inverse. Some examples are worked. After working through these materials, the student should be able

• to verify when the inverse of a differentiable function is differentiable;
• to find symbolically the derivative of an inverse function.
Modules:

• Recall the definition of inverse functions.

 Theorem. Suppose that f is a function which is differentiable on the open interval I. If either f '(x) > 0 or f '(x) < 0 for all x in I then f has an inverse f -1 which is defined and is differentiable on f(I). That is, for each a in I, f -1 is differentiable at b = f(a) and • Discussion [Using Flash]

• Examples:

• We record the results of the last example:

 Theorem.  • Derivatives of the inverse trigonometric functions.

• Drill problems on finding the derivatives of inverse functions.