We first consider the action of the Hasse derivatives on Drinfeld modular forms, which were shown by Uchino and Satoh to act as differential operators on the algebra of Drinfeld quasi-modular forms. While these operators do not preserve modularity, we show that they do preserve modularity modulo a prime ideal. We also study the behavior of the filtration under the action of the first Hasse derivative, and obtain results analogous to those obtained by Serre and Swinnerton-Dyer about Ramanujan's Theta-operator in the classical setting. We then consider a family of modular curves constructed by Drinfeld, and we study their Weierstrass points, a finite set of points of geometric interest. These curves are moduli spaces for Drinfeld modules with level structure, which are the objects which in our setting play a role analogous to that of elliptic curves. Previous work of Baker shows that for each Weierstrass point of these curves, the reduction modulo a certain prime ideal of the underlying Drinfeld module is supersingular. We study a modular form W for this congruence subgroup whose divisor is closely related to the set of Weierstrass points, an idea first presented by Rohrlich in the classical setting. To this end, we first establish a one-to-one correspondence between certain Drinfeld modular forms on the congruence subgroup and forms on the full modular group. In certain cases we can then use knowledge about the action of the Hasse derivatives to compute explicitly a form that is congruent to W modulo our prime ideal. This allows us to obtain an analogue of Rohrlich's result, which is the first important step towards obtaining a more precise relationship between the supersingular locus and Weierstrass points on our modular curves, as illustrated by Ahlgren and Ono in the classical setting.