Let
X
1
,
X
2
,
…
X_1,X_2,\ldots
be independent and identically distributed random variables in
C
{\mathbb {C}}
chosen from a probability measure
μ
\mu
and define the random polynomial
P
n
(
z
)
=
(
z
−
X
1
)
…
(
z
−
X
n
)
.
\begin{align*} P_n(z)=(z-X_1)\ldots (z-X_n)\,. \end{align*}
We show that for any sequence
k
=
k
(
n
)
k = k(n)
satisfying
k
≤
log
n
/
(
5
log
log
n
)
k \leq \log n / (5 \log \log n)
, the zeros of the
k
k
th derivative of
P
n
P_n
are asymptotically distributed according to the same measure
μ
\mu
. This extends work of Kabluchko, which proved the
k
=
1
k = 1
case, as well as Byun, Lee and Reddy [Trans. Amer. Math. Soc. 375, pp. 6311–6335] who proved the fixed
k
k
case.