Mollified moments of quadratic Dirichlet $L$-functions over function fields

We compute asymptotic formulae for the mollified first and second moments for the family of quadratic Dirichlet $L$-functions in the function field setting. As an application, we obtain non-vanishing results for the derivatives of the completed $L$-functions $\mathrm{\Lambda }(s,{\chi }_D)$ at the central point $s=1\mathrm{/}2$. In particular, we show that the proportion of $\Lambda^{(2k)}(\frac{1}{2},\chi_D) \neq 0$is$1+O(k^{-2})$as$k \to \infty$.