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A paper of Glasby, "The composition and derived lengths of a soluble group", 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 $d<\log_2\log_2|G|$ plus a (fairly small) constant (assuming $|G|>1$).