The Fr\"olicher-Nijenhuis bracket for derivation based non commutative differential forms

In commutative differential geometry the Fr"olicher-Nijenhuis bracket computes all kinds of curvatures and obstructions to integrability. In \cit!{3} the Fr"olicher-Nijenhuis bracket was developped for universal differential forms of non-commutative algebras, and several applications were given. In this paper this bracket and the Fr"olicher-Nijenhuis calculus will be developped for several kinds of differential graded algebras based on derivations, which were indroduced by \cit!{6}.