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We study the conditional upper bounds and extreme values of derivatives ofthe Riemann zeta function and Dirichlet $L$-functions near the 1-line. Let$\ell$ be a fixed natural number. We show that, if $|\sigma-1|\ll1/\log_2t$,then $|\zeta^{(\ell)}(\sigma+ it)|$ has the same maximal order (up to theleading coefficients) as $|\zeta^{(\ell)}(1+ it)|$ when $t\to\infty$. The range$1-\sigma\ll1/\log_2t$ is wide enough, since we also show that $(1-\sigma)\log_2t \to \infty\; (t \to \infty)$ implies$\limsup_{t\to\infty}|\zeta^{(\ell)}(\sigma+ it)| / (\log_2t)^{\ell+1} =\infty$. Similar results can be obtained for Dirichlet $L$-functions$L^{(\ell)}(\sigma,\chi)$ with $\chi\pmod q$.

On derivatives of zeta and $L$-functions near the 1-line