  In this paper, we obtain an analogue of the Serre derivation acting on the
product of spaces of Drinfeld modular forms which also generalizes the
differential operator introduced by Gekeler in the rank two case. We further
introduce a finitely generated algebra $\mathcal{M}_r$ containing all the
Drinfeld modular forms for the full modular group and show its stability under
the partial derivatives. Moreover, we obtain the transcendence of the non-zero
values of the generators of $\mathcal{M}_r$ and certain Drinfeld modular forms
at CM points over the rational function field.
