Show that if 0 is not a pole of $P/Q \in \mathbf{Q}(x)$, then there exists a unique power series with rational coefficients $g \in \mathbf{Q}\llbracket x \rrbracket$ such that $P = Q \cdot g$.
Show that the map $P/Q \longmapsto g$ is compatible with addition and multiplication in $\mathbf{Q}(x)$ and in $\mathbf{Q}\llbracket x \rrbracket$, and that it sends the derivative $(P/Q)' = (P'Q - PQ')/Q^2$ to the derived power series $g'$.