The derived series of a finite p-group

Let $G$ be a finite $p$-group, and let $\text{der}Gd$ denote the $d$-th term of the derived series of $G$. We show, for $p\ge 5$, that $\text{der}Gd\mathrm{\ne }1$ implies ${\log }_p\text{ord}G\ge 2^d+3d-6$, and hence we improve a recent result by Mann.