Derivations of the idealization of the incidence module over the finitary incidence algebra

Let $R$ be a unital commutative ring and $X$ a partially ordered set. Let $FI(X,R)$ and $I(X,R)$ be the finitary incidence algebra and the incidence module of $X$ over $R$, respectively, and let $D(X,R)=FI(X,R)\oplus I(X,R)$ be the idealization of the $FI(X,R)$-bimodule $I(X,R)$. We give a description of the Lie algebra of derivations of $D(X,R)$ and also provide some necessary or sufficient conditions on $R$ and $X$ for the description of this Lie algebra to be simpler.