Math Help - Derivative of a^x.

1. Derivative of a^x.

I am studying the attached image:

I find myself wondering is it:
$m(a)=\lim \Delta x\rightarrow 0\frac{a^\Delta^x-1}{\Delta x}$

or is it:

$m(a)=a^x\lim \Delta x\rightarrow 0\frac{a^\Delta^x-1}{\Delta x}$

2. Re: Derivative of a^x.

It is the first one.

3. Re: Derivative of a^x.

Why then is it said:

$m(a)=\frac{d}{dx}a^x ...at... x=0$

it is confusing me if m(a) is the first one I posted. Clearly before they said:

$m(a)=\frac{d}{dx}a^x=m(a)a^x$