We introduce the module of derivations \(\Theta _{h,M}\) attached to a given analytic map \(h:({\mathbb {C}}^n,0)\rightarrow ({\mathbb {C}}^p,0)\) and a submodule \(M\subseteq {\mathcal {O}}_n^p\) and analyse several exact sequences related to \(\Theta _{h,M}\) . Moreover, we obtain formulas for several numerical invariants associated to the pair (h, M) and a given analytic map germ \(f:({\mathbb {C}}^n,0)\rightarrow ({\mathbb {C}}^q,0)\) . In particular, if X is an analytic subvariety of \({\mathbb {C}}^n\) , we derive expressions for analytic invariants defined in terms of the module \(\Theta _X\) of logarithmic vector fields of X.
