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## Tuesday, 26 March 2013

### Differentiating Inverse Functions

Inverse functions are very important in Mathematics as well as in many applied areas of science. The most famous pair of functions inverse to each other are the logarithmic and the exponential functions. Other functions like the tangent and arctangent play also a major role.
In any case, let f(x) be an invertible function, with f-1(x) as its inverse, that is

May be the easiest way to remember this is to write

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. We will see in the coming pages that the logarithmic function  is the antiderivative of  for x > 0, with . This function has an inverse on , known to us as the exponential function ex. So this function is differentiable and

But we have , hence

This is truly an amazing result: the derivative of ex is the function itself. This property of the exponential function has many interesting applications.Example. Rational Powers. For x > 0, and any natural numbers n and m, we have

Let us first take care of the derivative of the function nthroot f(x)=x1/n, which is just the inverse function of xn. Clearly for x > 0, the derivative of xn is not 0, so f(x)=x1/n is differentiable and

Easy algebraic calculations give

In other words, the formula  is also valid for r=1/n, for  Back to our formula, to differentiate the function xn/m we will use the above results combined with the chain rule. So we have

Again this means that the formula  is valid even when r is a rational number. In particular combined with the chain rule, we get

for any differentiable function u(x).Example. We have

The following example discusses the above ideas for trigonometric functions.
Example. Consider the tangent function  on the interval . Then  has an inverse function  known as the arctangent function, which we prefer to denote by . Since

for any , then  is differentiable. Moreover we have

or

It might be surprising that the transcendental function  has as its derivative a rational function. This will be very useful once we try to integrate rational functions.