I know that $f(z)$ is analitic everywhere except of $z=0$, where it has a pole of order $k$.
Can I say that $f'(z)$ will have a pole of order $k+1$? For example, if $f(z)=\frac{1}{z^k} \rightarrow f'(z)=\frac{-k}{z^{k+1}}$, isn't it? Is that OK for every function $f(z)$?
What can I say about the residue of $f'(z)$ in $z=0$ and what can I say in $z=\infty$
$$Res[f'(z), z=0] = ???$$ $$Res[f'(z), z=\infty] = \frac{-1}{z^2}Res[f'(z), z=0] = ???$$
Thanks