Let $G$ be a nilpotent group of class $c$ and derived length $d$.
Then $d\le1+\log_2c$.
What we want to do is prove that this result can't be improved.
So I considered the group $C_p\wr C_p$ i.e. the wreath product of $C_p$ and $C_p$. We know that this group has nilpotence class equal to $p$.
Then I'm trying to prove that its derived length it's equal to $2$.
The only idea came to my mind is to use Ito Theorem (if a group $G$ can be written as $AB$ where $A$ and $B$ are abelian group, then $G^{(2)}=1$, i.e. $d\le2$); but if I really can take account of this theorem, who are the two abelian groups $A$ and $B$?