LetKbe a commutative ring, let ▵ be an abelian group, and let ϵ:▵x▵→Kbe a commutation factor over ▵.A ▵ gradedK-algebra is said to be ϵ-commutative if its ϵ-bracket is identically zero, (K,ϵ) derivations from a given ϵ-commutative ▵-gradedK-algebraAinto bimodules are studied. It is proved that for each λϵ▵ there exists a universal initial (k,ϵ)-derivation of degree λ ofA. For each λϵ▵ a natural module of (K, ϵ, λ)-differentials ofAalong with a differential map is constructed. It is proved that each derivation ofAcanonically equipps this module with a structure of differential module. Applications and examples are given. It is shown that the first order exterior differentials which are known from the theory of smooth graded manifolds are universal initial homogeneous derivations of the sort considered hereby.