Extreme values of derivatives of Dirichlet $L$-functions

In this paper, we establish lower bounds for extreme values of derivatives of Dirichlet $L$-functions in the range $1\mathrm{/}2<\sigma <1$. Compared with the work of Aistleitner, Mahatab, Munsch, and Peyrot in 2019, our result shows that derivatives of Dirichlet $L$-functions can attain extreme values of the same order of magnitude as the original $L$-functions when $1\mathrm{/}2<\sigma <1$ holds.