  In this series, we investigate the calculation of mean values of derivatives
of Dirichlet $L$-functions in function fields using the analogue of the
approximate functional equation and the Riemann Hypothesis for curves over
finite fields. The present paper generalizes the results obtained in the first
paper. For $\mu\geq1$ an integer, we compute the mean value of the $\mu$-th
derivative of quadratic Dirichlet $L$-functions over the rational function
field. We obtain the full polynomial in the asymptotic formulae for these mean
values where we can see the arithmetic dependence of the lower order terms that
appears in the asymptotic expansion.
