Derivative of $a^x$ from first principles I've been trying to think for the past few days how one could differentiate $a^x$ based on the definition that $a^n$ is repeated multiplication, $a^{n/m}=(\sqrt[m]a)^n$, and $a^x$ is the completion of the above function by continuity.
With a bit of algebra, the problem quickly reduces to finding the derivative at $0$:
$$\lim_{h\to0}\frac{a^h-1}{h}\tag{1}$$
And that limit's really got me stumped. Since you'll obviously have to use the definition in some way, I thought I'd replace $h$ with $\frac{1}{n}$ and use the $n$-th root definition:
$$\lim_{n\to\infty}n(\sqrt[n]a-1)\tag{2}$$
Of course, just because $(2)$ exists doesn't automatically imply $(1)$ exists, but it might be a first step. Even $(2)$ has me stumped, though.

Well, historically that is just the definition of the natural logarithm, so define
$$l(a)=\lim_{n\to\infty}n(\sqrt[n]{a}-1)$$
and conclude that this function has the usual properties of a logarithm, like $l(ab)=l(a)+l(b)$.
