I am trying to differentiate $2^x$ from first principles. This is what I have so far:
\begin{align} f'(x) &= \underset{h\rightarrow 0}{\textrm{lim}} \frac{f(x+h) - f(x)}{h}\\ \implies \frac{d2^x}{dx} &= \underset{h\rightarrow 0}{\textrm{lim}}\frac{2^{x+h}-2^x}{h}\\ &= \underset{h\rightarrow 0}{\textrm{lim}}\frac{2^x(2^h -1)}{h} \end{align}
From that point on, as the limit is of type 0/0, I was thinking of using L'Hôpital's rule, but this gives \begin{equation} \frac{d2^x}{dx} = 2^x\frac{d2^h}{dh}\bigg\rvert_{h=0}. \end{equation}
Not sure how to go from there.