Letbe a unital ring with a nontrivial idempotent, and letbe an-bimodule. We say that an additive mapis derivable atiffor anywith. In this paper, we give a necessary and sufficient condition for an additive mapto be derivable atwith. Moreover, we show that ifis a prime Banach algebra with the unit, then an additive mapis derivable atwithif and only if there is a derivationsuch thatfor all. As an application, we get a full characterization of derivable maps on some reflexive algebras and von Neumann algebras with no abelian summands. In particular, we show that an additive mapis derivable at any nonzero finite rank operator if and only if it is a derivation. New equivalent characterization of derivations on these algebras are obtained. [ABSTRACT FROM AUTHOR]