The concept was introduced by Schmidt & Hasse (1937).
For a (not necessarily commutative nor associative) ring B and a B-algebra A, a Hasse–Schmidt derivation is a map of B-algebras taking values in the ring of formal power series with coefficients in A.
This definition is found in several places, such as Gatto & Salehyan (2016, §3.4), which also contains the following example: for A being the ring of infinitely differentiable functions (defined on, say, Rn) and B=R, the map is a Hasse–Schmidt derivation, as follows from applying the Leibniz rule iteratedly.
Hazewinkel (2012) shows that a Hasse–Schmidt derivation is equivalent to an action of the bialgebra of noncommutative symmetric functions in countably many variables Z1, Z2, ...: the part
Basic properties of derivations in this context lead to a conceptual proof of the Cayley–Hamilton theorem.