 Solution

Let $\delta =4-D$ and $F(g)=g-C{g}^2/\delta +O(g^3)$. Since changing the arbitrary renormalization scale must leave the bare theory fixed,Write $\beta (g)=-\delta g+b{g}^2+O(g^3)$. Substitution gives, through order $g^2$,Therefore $b=-C$, and the specified residue gives the renormalization-group beta functionThe derivative is at fixed bare parameters. It does not say the physical theory changes when $\mu$ changes; the running coupling compensates the scale used to describe it. The coefficient also depends on defining the derivative with respect to $\log \mu$, rather than $\log {\mu }^2$; using the latter would divide both terms by two.