Derivatives of f(x)=a^x
Let's apply the definition of differentiation and see what happens:
![\begin{eqnarray*}
f^\prime(x_0) &\isdef & \lim_{\delta\to0} \frac{f(x_0+\delta)-...
...{\delta}
= a^{x_0}\lim_{\delta\to0} \frac{a^\delta-1}{\delta}.
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img283.png)
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
for
, it follows that there exists a
positive real number we'll call
such that for
we get
![$\displaystyle \lim_{\delta\to 0} \frac{e^\delta-1}{\delta} \isdef 1 .
$](http://www.dsprelated.com/josimages_new/mdft/img290.png)
![$ a=e$](http://www.dsprelated.com/josimages_new/mdft/img289.png)
![$ \left(a^x\right)^\prime =
(e^x)^\prime = e^x$](http://www.dsprelated.com/josimages_new/mdft/img291.png)
So far we have proved that the derivative of is
.
What about
for other values of
? The trick is to write it as
![$\displaystyle a^x = e^{\ln\left(a^x\right)}=e^{x\ln(a)}
$](http://www.dsprelated.com/josimages_new/mdft/img293.png)
![$ \ln(a)\isdef \log_e(a)$](http://www.dsprelated.com/josimages_new/mdft/img294.png)
![$ e$](http://www.dsprelated.com/josimages_new/mdft/img92.png)
![$ a$](http://www.dsprelated.com/josimages_new/mdft/img236.png)
![$\displaystyle \frac{d}{dx} f(g(x)) = f^\prime(g(x)) g^\prime(x)
$](http://www.dsprelated.com/josimages_new/mdft/img295.png)
![$ x_0$](http://www.dsprelated.com/josimages_new/mdft/img296.png)
![$\displaystyle \frac{d}{dx} f(g(x))\vert _{x=x_0} = f^\prime(g(x_0)) g^\prime(x_0).
$](http://www.dsprelated.com/josimages_new/mdft/img297.png)
![$ g(x)=x\ln(a)$](http://www.dsprelated.com/josimages_new/mdft/img298.png)
![$ g^\prime(x) = \ln(a)$](http://www.dsprelated.com/josimages_new/mdft/img299.png)
![$ f(y)=e^y$](http://www.dsprelated.com/josimages_new/mdft/img300.png)
![$ \left(a^x\right)^\prime = \left(e^{x\ln a}\right)^\prime
= e^{x\ln(a)}\ln(a) = a^x \ln(a)$](http://www.dsprelated.com/josimages_new/mdft/img301.png)
![$\displaystyle \zbox {\frac{d}{dx} a^x = a^x \ln(a).}
$](http://www.dsprelated.com/josimages_new/mdft/img302.png)
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