We are given a non-constant differentiable function $f(x)$ which satisfies the relation
$$\frac{f(x)}{f(y)}=f(x-y)$$
From the given data, how do I determine $f(x)$?
It is clear to me that the function should be of the form $a^{kx}$ but I can't really figure out how to come up with a rigorous proof. Also,is there any way to tell that $a^{kx}$ is in fact the only solution? I have tried substituting $x$ and $y$ with different values but I couldn't really get anywhere.