4.3 derivatives of inv erse trig. functions

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