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## Derivatives of f(x)=a^x

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.,