  It is proved that if $T$ is sufficiently large, then uniformly for all
positive integers $\ell \leqslant (\log T) / (\log_2 T)$, we have
\begin{equation*} \max_{T\leqslant t\leqslant
2T}\left|\zeta^{(\ell)}\Big(1+it\Big)\right| \geqslant e^{\gamma}\cdot
\ell^{\ell}\cdot (\ell+1)^{ -(\ell+1)}\cdot\Big(\log_2 T - \log_3 T +
O(1)\Big)^{\ell+1} \,, \end{equation*} where $\gamma$ is the Euler constant. We
also establish lower bounds for maximum of
$\big|\zeta^{(\ell)}(\sigma+it)\big|$ when $\ell \in \mathbb N $
  and $\sigma \in [1/2, \,1)$ are fixed.
