  Let $x_1, \dots, x_n$ be $n$ independent and identically distributed random
variables with mean zero, unit variance, and finite moments of all remaining
orders. We study the random polynomial $p_n$ having roots at $x_1, \dots, x_n$.
We prove that for $\ell \in \mathbb{N}$ fixed as $n \rightarrow \infty$, the
$(n-\ell)-$th derivative of $p_n^{}$ behaves like a Hermite polynomial: for $x$
in a compact interval,$${n^{\ell/2}} \frac{\ell!}{n!} \cdot
p_n^{(n-\ell)}\left( \frac{x}{\sqrt{n}}\right) \rightarrow He_{\ell}(x +
\gamma_n),$$ where $He_{\ell}$ is the $\ell-$th probabilists' Hermite
polynomial and $\gamma_n$ is a random variable converging to the standard
$\mathcal{N}(0,1)$ Gaussian as $n \rightarrow \infty$. Thus, there is a
universality phenomenon when differentiating a random polynomial many times:
the remaining roots follow a Wigner semicircle distribution.
