Let A be a Banach algebra with unity I containing a non-trivial idempotent P and M be a unital A-bimodule. Under several conditions on A, M and P, we show that if d is an additive mapping from A into M satisfying d(ab)=ad(b)+d(a)b for any a,b in A with ab=P, then d is a derivation or d(a)=D(a)+an for some additive derivation D from A into M and some n in M . As applications of above results, we characterize the additive mappings derivable at P on matrix algebras, Banach space nest algebras, standard operator algebras and nest subalgebras of von Neumann algebras. Moreover we obtain some results about automatic continuity of linear (additiv mappings) derivable at P on various Banach algebras
