for $f(z)$ Meromorphic we know that $f(z)$ on $\Omega$ - an open set ,is Holomorphic on $\Omega/A$ when $A$ is discrete set of the poles of $f(z)$.
I know that $f'(z)$ is also Holomorphic on $\Omega/A$ because $f(z)$ analytic on this set. Would it be enough to write $f(z)=\frac {g(z)}{h(z)}$ and $g(z),h(z)$ are holomorphic functions on $\Bbb C$, and then $f'(z)=\frac {g'(z)h(z)-g(z)h'(z)}{(h(z))^2}$ so we know that the set of zeros of $f(z), f'(z)$ denominator is the same, so we know this set is A and it's elements are poles for $f(z)$ and $f'(z)$?
Yes, that would be enough. When you differentiate a meromorphic function, your are not adding any new poles, and the original poels remain poles (that is, they don't become essential singularities).