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In this short note, we study the derivatives of all orders for the randomfield $$ X_T(h) = \sum_{p \leq T} \frac{\text{Re}(U_p \, p^{-i h})}{p^{1/2}},\quad h\in [0,1], $$ where $(U_p, \, p ~\text{primes})$ is an i.i.d. sequenceof uniform random variables on the unit circle in $\mathbb{C}$. We show thatthe maximum of $X_{T}$, and more generally the maximum of its $j$-thderivative, varies on a $(\log T)^{-\frac{1}{2}(j+2)}$ scale, which improvesand extends the main result in Arguin & Ouimet (2019) and makes furtherprogress towards the open problem of the tightness of the recentered maximum of$X_{T}$. Our proof is also much simpler and shorter.

Discretization of the maximum for the derivatives of a random model of $\log|\zeta|$ on the critical line