Let $\mathcal{N}$ be a nest on a Banach space $\mathcal{X}$ and, suppose that $\mathcal{A}$ is a subalgebra of $\mathcal{B}(\mathcal{X})$ contains all rank one operators in $Alg\mathcal{N}$ and identity operator $I$, which is a Banach algebra with respect to some norm. Let there exists a non-trivial idempotent $P\in \mathcal{A}$ with $P(\mathcal{X})\in \mathcal{N}$. We show that if $d:\mathcal{A} \rightarrow \mathcal{B}(\mathcal{X})$ 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. As applications of the above result, we characterize the additive mappings derivable at $P$ on Banach space nest algebras and standard operator algebras.
