I've tried for a while myself from first principles and applying various rules, but always end up going in circles. I've gotten as far as
$$ y = a^x $$ $$ \frac{dy}{dx} = a^x \left( \lim_{x \rightarrow 0} \frac{a^h-1}{h} \right) $$
but I have no idea how I should go about cancelling the $h$ in the denominator. Any help is appreciated.