  Let $X_1,X_2,\ldots$ be independent and identically distributed random
variables in $\mathbb{C}$ chosen from a probability measure $\mu$ and define
the random polynomial $$ P_n(z)=(z-X_1)\ldots(z-X_n)\,. $$ We show that for any
sequence $k = k(n)$ satisfying $k \leq \log n / (5 \log\log n)$, the zeros of
the $k$th derivative of $P_n$ are asymptotically distributed according to the
same measure $\mu$. This extends work of Kabluchko, which proved the $k = 1$
case, as well as Byun, Lee and Reddy who proved the fixed $k$ case.
