How do we show that the function which is its own derivative is exponential?
Hint. If $f$ is such a function then calculating the derivative of $f(x)/e^x$ provides useful information about $f$.
If $f'=f$, take $g(x)=f(x)e^{-x}$. Then $g'(x)=f'(x)e^{-x}-f(x)e^{-x}=0$ and so $g$ is constant. The constant is $g(0)=f(0)$. Therefore, $f'=f$ implies $f(x)=f(0)e^{x}$.