Differentiation — finding the derivative of an inverse function

Martin McBride
7 min readApr 3, 2024

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):

Of course, if g is the inverse of f, it follows that f is the inverse of g:

Another way to express this is to say that g undoes f, and vice versa. Notice that, for the natural log, f can only be applied to x > 0 (because we can’t find the log of a non-positive number). So:

Now let’s find the slope of f(x) at some point A. This can be found by drawing a right-angled triangle that forms a tangent to the curve at A.