The nondifferentiability of the solution operator of the obstacle problem is the main challenge in tackling optimization problems with an obstacle problem as a constraint. Therefore, the structure of the Bouligand generalized differential of this operator is interesting from a theoretical and from a numerical point of view. The goal of this article is to characterize and compute a specific element of the Bouligand generalized differential for a wide class of obstacle problems. We allow right-hand sides that can be relatively sparse in H-1(Omega) and do not need to be distributed among all of H-1(Omega). Under assumptions on the involved order structures, we investigate the relevant set-valued maps and characterize the limit of a sequence of Gateaux derivatives. With the help of generalizations of Rademacher's theorem, this limit is shown to be in the considered Bouligand generalized differential. The resulting generalized derivative is the solution operator of a Dirichlet problem on a quasi-open domain.