How to prove that if $ f(x) = f'(x) \Rightarrow f(x) = c e^x$

Question

In calculus, If I know that a derivative of a function is the function itself, how do I prove that the function equals $e^x$:

$$ f(x) = f'(x) \Rightarrow f(x) = ce^x $$ Why is this always true?

Thank you