  We obtain the asymptotic main term of moments of arbitrary derivatives of
$L$-functions in the function field setting. Specifically, the first, second,
and mixed fourth moments. The average is taken over all non-trivial characters
of a prime modulus $Q \in \mathbb{F}_q [t]$, and the asymptotic limit is as
$\mathrm{deg} \, Q \longrightarrow \infty$. This extends the work of Tamam who
obtained the asymptotic main term of low moments of $L$-functions, without
derivatives, in the function field setting. It also expands on the work of
Conrey, Rubinstein, and Snaith who cojectured, using random matrix theory, the
asymptotic main term of any even moment of the derivative of the Riemann
zeta-function in the number field setting.
