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Let A be an associative algebra (or any other kind of algebra for thatmatter). A derivation on A is an endomorphism \del of the underlying Abeliangroup of A such that \del(ab)=a(\del b)+(\del a)b for all a,b\in A (1.1)  A Hasse-Schmidt derivation is a sequence (d_0=id,d_1,d_2,...,d_n,...) ofendomorphisms of the underlying Abelian group such that for all n \ge 1d_n(ab)= \sum_{i=0}^n (d_ia)(d_{n-i}b) (1.2)  Note that d_1 is a derivation as defined by (1.1). The individual d_n thatoccur in a Hasse-Schmidt derivation are also sometimes called higherderivations. A question of some importance is whether Hasse-Schmidt derivationscan be written down in terms of polynomials in ordinary derivations. Forinstance in connection with automatic continuity for Hasse-Schmidt derivationson Banach algebras. Such formulas have been written down by, for instance,Heerema and Mirzavaziri in [5] and [6]. They also will be explicitly givenbelow. It is the purpose of this short note to show that such formulas followdirectly from some easy results about the Hopf algebra NSymm of non-commutativesymmetric functions. In fact this Hopf algebra constitutes a universal exampleconcerning the matter.

Hasse-Schmidt derivations and the Hopf algebra of noncommutative symmetric functions