Differentiation — finding the derivative of an inverse function

Often if we are trying to differentiate a difficult function, we can make progress by finding the derivative of a related function, and then using the relationship to find the derivative of the original function.

In this article, we will look at one such technique. If we find it difficult to differentiate a function f(x), it might be easier to differentiate its inverse function, g(x). Having found g’(x) we might then be able to find the result we need, f’(x).

As a first example, we will find the derivative of the natural logarithm function. We will then generalise that to any function, and prove the result.

Finding a derivative using the inverse function

Here is our initial function f(x). It is the natural logarithm function, but we will call it f(x) for now:

We will make use of the inverse function. As you may know, the inverse of the natural log is the exponential function, e to the power x. But for now, we will call it g(x):