This is a follow-up to this question. The one thing that did not get a completely satisfactory answer there is: If $\Phi_n$ denotes the $n$-th cyclotomic polynomial, and $\zeta^k_n = e^{2k\pi i/n}$ is one of its (complex) roots, is there are nice closed formula for the value of the derivative at the root,

$\Phi'_n(\zeta^k_n) = ?$

"Nice" is of course subjective; what I hope for is a closed formula, depending on $n$ and $k$, for the real and imaginary part as some rational function in $\cos(2k\pi/n)$ and $\sin(2k\pi/n)$.

Ideas so far: Algebraically, one easily gets the formulae (for any root $\zeta$ of $\Phi_n$):

$$(1) \qquad \qquad\displaystyle \Phi'_n(\zeta) = \prod^{n-1}_{\quad i=2\\ \gcd(i,n)=1} (\zeta - \zeta^i)$$

as well as (idea from Jyrki Lahtonen's comment to the original question):

$$(2) \qquad \qquad \quad \displaystyle \Phi'_n(\zeta) = \dfrac{n}{\zeta \cdot \displaystyle \prod_{d | n\\ d\neq n} \Phi_d (\zeta)}$$

For prime $n =p$, formula (2) reduces to the seemingly easy special case

$$\displaystyle \Phi'_p(\zeta) = \dfrac{p}{\zeta \cdot (\zeta-1)}.$$

Now one can put the complex number into that; I've played around a bit, but fail to see a general pattern.


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