  It has been shown that zeros of Kac polynomials $K_n(z)$ of degree $n$
cluster asymptotically near the unit circle as $n\to\infty$ under some
assumptions. This property remains unchanged for the $l$-th derivative of the
Kac polynomials $K^{(l)}_n(z)$ for any fixed order $l$. So it's natural to
study the situation when the number of the derivatives we take depends on $n$,
i.e., $l=N_n$. We will show that the limiting global behavior of zeros of
$K_n^{(N_n)}(z)$ depends on the limit of the ratio $N_n/n$. In particular, we
prove that when the limit of the ratio is strictly positive, the property of
the uniform clustering around the unit circle fails; when the ratio is close to
1, the zeros have some rescaling phenomenon. Then we study such problem for
random polynomials with more general coefficients. But things, especially the
rescaling phenomenon, become very complicated for the general case when
$N_n/n\to 1$, where we compute the case of the random elliptic polynomials to
illustrate this.
