  We show that the Drinfeld modular forms with $A$-expansion that have been
constructed by the author are precisely the hyperderivatives of the subfamily
of single-cuspidal Drinfeld modular forms with $A$-expansions that remain
modular after hyperdifferentiation. In addition, we show that
Drinfeld-Poincar\'{e} series display a similar behavior with respect to
hyperdifferentiation, giving indirect evidence that the Drinfeld modular forms
with $A$-expansions are Drinfeld-Poincar\'{e} series. The Drinfeld-Poincar\'{e}
series that we consider generalize previous examples of such series by Gekeler,
and Gerritzen and van der Put.
