If $f$ is a non-constant elliptic function, then $z\mapsto f'(z)/f(z)$ is a non-constant elliptic function

Question

How can I prove

If $f$ is a non-constant elliptic function, then $z\mapsto f'(z)/f(z)$ is a non-constant elliptic function?

To prove that $z\mapsto f'(z)/f(z)$ is elliptic, it suffices to notice that $$\frac{f'(z+\omega)}{f(z+\omega)}=\frac{f'(z)}{f(z)}$$ for any period $\omega$ of $f$.

But how can I prove that $f'(z)/f(z)$ is not a constant?

I observed that it fails for some periodic non-elliptic functions, for example if we choose $f=\exp$ then $f'(z)/f(z)$ is a constant.

Answer 1

If $f'/f$ is constant, then $f(z)=ke^{cz}$. Because every elliptic function free of poles must be constant, we conclude $c=0$ and $f(z)=k$. This proves the contrapositive version of the statement.