  In this paper we survey the notion and basic results on multivariate
Hasse--Schmidt derivations over arbitrary commutative algebras and we associate
to such an object a family of classical derivations. We study the behavior of
these derivations under the action of substitution maps and we prove that, in
characteristic $0$, the original multivariate Hasse--Schmidt derivation can be
recovered from the associated family of classical derivations. Our
constructions generalize a previous one by M. Mirzavaziri in the case of a base
field of characteristic $0$.
