The [[derivative]] of an exponential function is the same as the function itself. To demonstrate, let's find the derivative of $f(x) = e^x$ using the definition of the derivative. First we can find $f(x + h)$ $f(x + h) = e^{x+h}$ Plug this in to the definition of the derivative $f'(x) = \lim_{h \to 0}\frac{e^{x+h} - e^x}{h}$ Noticing that $e^{x+h} = e^xe^h$, we get $f'(x) = \lim_{h \to 0}\frac{e^xe^h - e^x}{h} = \lim_{h \to 0}\frac{e^x(e^h - 1)}{h}$ We can factor the term $e^x$ out of the limit as it doesn't depend on $h$. The term $\frac{e^h-1}{h}$ as $h$ approaches 0 is $1$. Thus we get that the limit of this exponential function is the same as the function itself! $f'(x) = e^x$ We will see that this result holds for all exponential functions. The derivative of $e^{ax}$ is $\frac1a e^{ax}$ . More generally, divide $e^{ax}$ by the derivative of $a$.