TITLE: Derivatives of the nth cyclotomic polynomial
QUESTION [15 upvotes]: Are there any useful properties of the $k$th derivative of the $n$th cyclotomic polynomial? 
In particular, what would the value of this be at $1$ and $0$, or any properties of the value of the $k$th derivative at $0$ or $1$?

REPLY [5 votes]: The derivative $\Phi_n'(t)$ is useful not only for $t=0$ or $t=1$, but for $t=\zeta_{p^r}$, a $p^r$-th primitive root of unity. To show that of all prime ideals in $\mathbb{Z}$, only $P=(p)$ is ramified in the cyclotomic field $K=\mathbb{Q}(\zeta_{p^r})$, one needs that the discriminant is given by
$$
D(1,\zeta_{p^r},\ldots ,\zeta_{p^r}^{\phi(p^r)})=\pm N_{K/\mathbb{Q}}(\Phi_n'(\zeta_{p^r}))=\pm p^{rp^r-rp^{r-1}-p^{r-1}}.
$$