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