 # Differentiating Inverse Functions

## Inverse Function Review

One application of the chain rule is to compute the derivative of an inverse function. First, let's review the definition of an inverse function:

We say that the function is invertible on an interval [a, b] if there are no pairs in the interval such that and . That means there are no two x-values that have the same y-value. That's important, because if two x-coordinates map to the same y-coordinate, the inverse function (working in reverse) would map a single x-coordinate to multiple y-coordinates. That doesn't make sense, because f(x) could have more than one resulting value!

We say that is the inverse of an invertible function on [a, b] if: For example, the functions and are inverses on since on that interval. Note that it works both ways -- the inverse function of the original function returns x, and the original function performed on the inverse ALSO returns x.

### Taking The Derivative

So, how do we differentiate an inverse function? Recall the chain rule: Applying this to the definition of an inverse function, we have: So: Let's see how to apply this to real examples.

#### Example 1

Let so as above. Then , and applying the formula we have:  This agrees with the answer we would get from viewing as the polynomial function .

#### Example 2

The function is invertible on the interval , with inverse . We know that , so applying our formula we see that We can check that , which means that .