The main goal of this article is introducing a fairly general framework to treat several types of derivations simultaneously. Let A and B be Ba- nach algebras, α and β be homomorphisms from A onto B, and X be a Banach B-bimodule. A map D ∈ B(A,X) is called an (α, β)-derivation if D(ab) = α(a)D(b) + D(a)β(b). All homomorphisms, ordinary deriva- tions, skew derivations, and point derivations are certain types of (α, β)- derivations. We define (α, β)-analogue of notions of amenability and weak amenability. Amenability modulo a closed ideal in the sense of [26] and generalized weak amenability in the sense of [2] are special cases of our concepts. We provide several characterizations of (α, β)-weak amenabil- ity under some mild conditions. Moreover (α∗∗, β∗∗)-weak amenability of A∗∗ implies (α, β)-weak amenability of A.
