If $f_n:U\to\mathbb{C}$ are holomorphic and $f=\lim_{n\to\infty} f_n$ uniformly on each compact subset of $U$, then $f'$ has no zeros, provided none of the $f'_n$'s have any and $f$ is not constant. Indeed, then $f'_n\to f'$ uniformly on all compact subsets of $U$ and if $f'(z)=0$, then we can apply Rouch\'e's theorem to $f',f'-f'_n$ and a disk $D\subset U$ such that $z\in D$, $f'$ has no zeros on $\partial D$ and $sup_{\partial D}|f'-f_n'|<inf_{\partial D}|f'|$ to conclude that $f'_n$ has a zero in the interior of $D$ as well.
On the other hand, it may happen that $f$ is constant: take $U$ to be the unit disk and $f_n(z)=z/n$.
