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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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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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 $(a^\ as $\ is less than 1 for $a=2$ and greater than $1$ for $a=3$ (as one can show via direct calculations), and since $(a^\ is a continuous function of$a$, it follows that there exists a positive real number we’ll call $e$such that for $a=e$ we get
\

For $a=e$, we thus have $\.

So far we have proved that the derivative of $e^x$ is $e^x$. What about $a^x$ for other values of $a$? 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, $g(x)=x\ so that $g^\, and $f(y)=e^y$ which is its own derivative. The end result is then $\, i.e.,
\

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