Q:

Why does $\frac{d}{dx} a^x = \ln(a)a^x$ and the derivative of $x^n$ equals $nx^{x-1}$

On mathisfun.com I found a list of derivative rules that seem to contradict each other. I'm probably missing something, but under "common functions" it says a function of $a^x$ has a derivative of $\ln(a)a^x$. This seems to contradict with another derivative rule that they called the "Power Rule", where $x^n$ is supposed to have a derivative of $nx^{x-1}$.
I am not very good at math. Could someone please explain why both of these statements are apparently true?

A:

The derivative of $a^x$ comes from the definition of $a^x$:
$$a^x\overset{\text{def}}{=}\mathrm e^{x\ln a},$$
so applying the chain rule, you obtain
$$(a^x)=(\mathrm e^{x\ln a})'\cdot (x\ln a)'= \mathrm e^{x\ln a}\cdot\ln a=a^x\cdot \ln a $$

A:

That's because the power rule only applies when the $x$ is in the bottom, for example
$$x^2, x^{1.5}, x^{1/3}, x^{-1}, x^{-\pi}$$
All those functions can be differentiated using power rule. 
However, exponential functions, those with $x$ in the top, like the following, can not be differentiated using power rule:
$$2^x, e^{-x}, x^x, \frac 1{0.3^x}$$

For more detail, the derivative formula for power functions is just
  $$\lim_{h\to 0}\frac{(x+h)^a-x^a}{h}$$
  This is just a powers of $x$ and $h$ multiplied together, while the derivative formula for exponential functions is 
  $$\lim_{h\to 0}\frac{a^{x+h}-a^x}{h}=\lim_{h\to 0}\frac{a^{x}a^h-a^x}{h}=a^{x}\lim_{h\to 0}\frac{a^h-1}{h}$$
  which looks and behaves very differently from the power series.
  Evaluating these limits gets a little hard, and people do have new tools and definitions that they use to help with that, but for now, I think this explanation covers the foundations.

