I am reading An Introduction to Knot Theory by W.B. Raymond Lickorish. In Chapter 11 the motivation for the Fox derivative is mentioned. I understand why the contribution of the occurrence of $x_j$ in $r_i = w_1x_jw_2$ is $\alpha(w_1)\tilde{x}_j$, but I do not understand why that for $x_j^{-1}$ in $r_i = v_1x_j^{-1}v_2$ is $-\alpha(v_1x_j^{-1})\tilde{x}_j$ and not $-\alpha(v_1)\tilde{x}_j$.
For people not having read the book, let me explain what the notations mean: Let the fundamental group of the complement $X$ of a knot be $G = <x_1, x_2, ...,x_n; r_1, r_2, ..., r_m>$, and then a complex P with $0$-cell $V$, the $1$-cells ${x_i}$ with endpoints identified with $V$, and ${c_i}$ be $2$-cells with boundaries $\partial c_i$ glued to the one cells ${x_i}$ according to the words ${r_i}$. Let $\tilde{P}$ be the infinite cyclic cover of $P$. Let $\tilde{V}$, ${\tilde{x_i}}$ and ${\tilde{c_i}}$ be the respective chosen lifts of $V$, ${x_i}$ that starts at $\tilde{V}$ and ${c_i}$ that starts at $\tilde{V}$.
Let $\alpha : G \rightarrow G / [G,G] = <t>$ be the abelianizer map. We would like to consider what contribution $x_j^{-1}$ makes to $\partial_2(\tilde{c_i})$ when it occurs in the relator $r_i$.