Let $\mathcal{A}$ be a Banach algebra with unity $I$ containing a non-trivial idempotent $P$ and $\mathcal{M}$ be a unital $\mathcal{A}$-bimodule. Under several conditions on $\mathcal{A}$, $\mathcal{M}$ and $P$, we show that if $d:\mathcal{A} \rightarrow \mathcal{M}$ is an additive mapping derivable at $P$ (i.e. $d(AB)=Ad(B)+d(A)B$ for any $A,B\in \mathcal{A}$ with $AB=P$), then $d$ is a derivation or $d(A)=\tau(A)+AN$ for some additive derivation $\tau:\mathcal{A} \rightarrow \mathcal{M}$ and some $N\in \mathcal{M}$, and various examples are given which illustrate limitations on extending some of the theory developed. Also, we describe the additive mappings derivable at $P$ on semiprime Banach algebras and $C^{*}$-algebras. 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 (additive) mappings derivable at $P$ on various Banach algebras.
