Summary: Let be a Banach algebra and let and be continuous homomorphisms on . We consider the following module actions on ,
We denote by the above -module. We call the Banach algebra -weakly amenable if every derivation from into is inner. In this paper, among other things, we investigate the relations between weak amenability and -weak amenability of . Some conditions can be imposed on such that the -weak amenability of implies the -weak amenability of .