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
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.