  This paper presents a cohomological study of modified Rota-Baxter associative
algebras in the presence of derivations. The Modified Rota-Baxter operator,
which is a modified version and closely related to the classical Rota-Baxter
operator, has garnered significant attention due to its applications in various
mathematical and physical contexts. In this study, we define a cohomology
theory and also investigate a one-parameter formal deformation theory and
abelian extensions of modified Rota-Baxter associative algebras under the
influence of derivations.
