A paper of Glasby, "[The composition and derived lengths of a soluble group](https://doi.org/10.1016/0021-8693(89)90204-4)", shows that if a soluble group $G$ has composition length $n$ then its derived length $d$ satisfies $d < 3 \log_2(n) + 9$. Since $n \leqslant \log_2|G|$, this makes $d<3\log_2\log_2|G|+9$ (assuming $|G|>1$).