If $f'(z)$ has an essential singularity can we say that $f(z)$ has an essential singularity at $z_0$.
Let $$\sum_{n=-\infty}^{\infty}a_n (z-z_0)^n$$ be the Laurent series of $f(z)$ near $z_0$. Then the Laurent series of $f'(z)$ near $z_0$ is $$\sum_{n=-\infty}^{\infty}n a_n (z-z_0)^{n-1}$$ Since this point is an essential singularity, there must be infinitely many non-zero coefficients $n a_n$ for negative $n$. Now what can we say about our original Laurent series of $f$?