# Derivatives of f(x)=a^x

## GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Derivatives of f(x)=a^x

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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., << Previous page  TOC  INDEX  Next page >>