Derivatives of Inverse

Trigonometric Function

Inverse Trig Functions

Some trig functions domains’ have to be restricted in order for them to have an inverse function – why?

Only functions that are 1-to-1 can have inverse functions

Find If

therefore

One more example

Find if

Differentiability of Inverse Functions

If f(x) is differentiable on an interval I, one may wonder whether f-1(x) is also differentiable? The answer to this question hinges on f'(x) being equal to 0 or not . Indeed, if for any , then f-1(x) is also differentiable. Moreover we have

Using Leibniz's notation, the above formula becomes

which is easy to remember.

Example:Confirm Differentiability of Inverse Function formula for the function

Solution:

and

MONOTONIC FUNCTIONS:Suppose that the domain of a function f is on an open interval I on which f’(x) > 0 or on which f’(x) < 0. Then f is one-to-one, f-1(x) is differentiable at all values of x in the range of f.

Example:

Consider the function .Show that f(x) is one-to=one function.

Solution:

Since f’(x) > 0 on the entire domain, f(x) is monotonic, therefore it has an inverse