Dependence of the derived $p$-length of a $p$-solvable group on the order of its Sylow $p$-subgroup

It is proved that the derived $p$-length $l_p^a(G)$ of the $p$-solvable group $G$ in which the Sylow $p$-subgroup has order $p^n$ is at most $1+\frac n2$ and if $p\not\in\{2,3\}$ then $l_p^a(G)\leqslant\frac{n+1}2$.