Let $R$ be a (commutative or non-commutative) unital ring, $X$ be a non-empty set, and $R \langle\! \langle X \rangle\! \rangle$ be the ordered series ring (in fact, a ring of formal power series over $R$ in $|X|$ non-commuting variables) obtained by endowing the set of all functions $\mathscr F(X) \to R$ with the usual operations of pointwise addition and Cauchy product. Here, $\mathscr F(X)$ is the free monoid on $X$, whose operation (that is, word concatenation) I'll denote by $\ast$.

While looking for a counterexample to a certain property in the class of local rings, I happened to note (with the help of Daniel Smertnig, who pointed out a problem with my previous formula) that, for each $z \in X$, the mapping $\partial_z$ that sends an ordered series $f \in R \langle\! \langle X \rangle\! \rangle$ to the function $$ \mathscr F(X) \to R \colon \mathfrak z \mapsto f(\mathfrak z) {\sum}_{(\mathfrak u, \mathfrak v) \in \mathscr F(X) \times \mathscr F(X): \mathfrak u \ast z \ast \mathfrak v = \mathfrak z} \delta_{u \ast v}, $$ is a well-defined derivation of $R \langle\! \langle X \rangle\! \rangle$ (here, $\delta_{u \ast v}$ means the Kronecker delta centered at the $X$-word $u \ast v$). In particular, the Leibniz identity follows from the fact that $\mathscr F(X)$ is a cancellative monoid with trivial group of units and every $X$-word factors uniquely in $\mathscr F(X)$ as a product of elements of $X$ (that is, $X$-words of length one).

My question is **whether anyone here can offer a reference** where $\partial_z$ is being introduced: I thought I would have found $\partial_z$ defined in Cohn's book on FIRs (where ordered series rings are discussed in Sect. 1.5), but it's not there (as far as I can see). I've also tried with Lam's *A First Course in Noncommutative Rings*, but the conclusion is the same.

Noncommutative rational series and applications(I was addressed to the same book by Daniel Smertnig, though for a different reason), and there doesn't seem to be much about derivations in it. In any case, I'll try to write to Reutenauer. $\endgroup$