Let’s apply the definition of differentiation and see what happens:
Since the limit of as is less than 1 for and greater than for (as one can show via direct calculations), and since is a continuous function of, it follows that there exists a positive real number we’ll call such that for we get
For , we thus have .
So far we have proved that the derivative of is . What about for other values of ? The trick is to write it as
and use the chain rule. Formally, the chain rule tells us how do differentiate a function of a function as follows:
In this case, so that , and which is its own derivative. The end result is then , i.e.,