Let $\mathcal L$ be an algebra generated by the commuting independent nests, $\mathcal M$ is an ultra-weakly closed subalgebra of $\mathbf B(\mathbf H)$ which contains $alg\mathcal L$ and $\phi$ is a norm continuous linear mapping from $alg\mathcal L$ into $\mathcal M$. In this paper we will show that a norm continuous linear derivable mapping at zero point from $Alg\mathcal L$ to $\mathcal M$ is a derivation.