**Proof.**Let f(x) = e^x. We will be using the first principle derivative:

\begin{align*} f'(x) &= \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h} \\ &= \lim_{h \rightarrow 0} \frac{e^{x+h} - e^x}{h} \\ &= \lim_{h \rightarrow 0} \frac{e^x(e^h - 1)}{h} \\ &= e^x \cdot \lim_{h \rightarrow 0} \frac{e^h - 1}{h}. \end{align*}

\begin{align*} f'(x) = e^x \cdot \lim_{h \rightarrow 0} \frac{e^h - 1}{h} = e^x, \end{align*}