Let $Q \in \mathbb{C}[X]$ be a non-zero polynomial. We define the function:
$$\begin{aligned} \varphi : [0,+\infty[ &\rightarrow \mathbb{R} \\ p &\mapsto \begin{cases} \ln\left(M_p(Q)\right) & \text{if } p > 0 \\ 0 & \text{if } p = 0 \end{cases} \end{aligned}$$
where $M_p(Q) = \frac{1}{2\pi}\int_0^{2\pi}\left|Q(e^{i\theta})\right|^p d\theta$. Show carefully that $\varphi$ is differentiable on $]0,+\infty[$ and compute its derivative on this interval.