Superpositions of free Fox derivations

Fox derivations are an effective tool for studying free groups and their group rings. Let $F_r$ be a free group of finite rank $r$ with basis $\{f_1, \ldots, f_r\}.$ For every $i$, the partial Fox derivations $\partial /\partial f_i$ and $\partial /\partial f_i^{-1}$ are defined on the group ring $\mathbb{Z}[F_r]$. For $k\geq 2$, their superpositions $D_{f_i^{\epsilon}} = \partial /\partial f_i^{\epsilon_k} \circ \ldots \circ \partial /\partial f_i^{\epsilon_1}, \epsilon = (\epsilon_1, \ldots , \epsilon_k) \in \{\pm 1\}^k,$ are not Fox derivations. In this paper, we study the properties of superpositions $D_{f_i^{\epsilon}}$. It is shown that the restrictions of such superpositions to the commutant $F_r'$ are Fox derivations. As an application of the obtained results, it is established that for any rational subset $R$ of $F_r'$ and any $i$ there are parameters $k$ and $\epsilon$ such that $R$ is annihilated by $D_{f_i^{\epsilon}}$.