### Derivative of Exponential function

Let $y=f\left(x\right)={e}^{x}$, then $\frac{dy}{dx}=\underset{h\to 0}{\mathrm{lim}}\frac{f\left(x+h\right)-f\left(x\right)}{h}$

$=\underset{h\to 0}{\mathrm{lim}}\frac{{e}^{x+h}-{e}^{x}}{h}$

$=\underset{h\to 0}{\mathrm{lim}}\frac{{e}^{x}{e}^{h}-{e}^{x}}{h}$

$=\underset{h\to 0}{\mathrm{lim}}\frac{{e}^{x}\left({e}^{h}-1\right)}{h}$

$=\underset{h\to 0}{\mathrm{lim}}{e}^{x}\frac{\left({e}^{h}-1\right)}{h}$

$=\left({e}^{x}\right)\underset{h\to 0}{\mathrm{lim}}\frac{{e}^{h}-1}{h}$

$=\left({e}^{x}\right)\left(1\right)$

$={e}^{x}$

In the above, we have used the fact that $\underset{h\to 0}{\mathrm{lim}}\frac{{e}^{h}-1}{h}=1$