  A LieDer pair (respectively, an AssDer pair) is a Lie algebra equipped with a
derivation (respectively, an associative algebra equipped with a derivation). A
couple of LieDer pair structures on a vector space are called Compatible LieDer
pairs (respectively, compatible AssDer pairs) if any linear combination of the
underlying structure maps is still a LieDer pair (respectively, AssDer pair)
structure. In this paper, we study compatible AssDer pairs, compatible LieDer
pairs, and their cohomologies. We also discuss about other compatible
structures such as compatible dendriform algebras with derivations, compatible
zinbiel algebras with derivations, and compatible pre-LieDer pairs. We describe
a relationship amongst these compatible structures using specific tools like
Rota-Baxter operators, endomorphism operators, and the commutator bracket.
