I'm studying these PDF notes on strings on curved backgrounds and the author introduces the dimensional regularization of the theory by first defining the bare and renormalized target space metric, $G_{\mu \nu}^\epsilon, \, G_{\mu \nu}$, respectively by
$$G_{\mu \nu}^\epsilon = \mu^{-\epsilon}\left(G_{\mu \nu} \ + \ \sum_{n \ge 1} \frac{K^{(n)}_{\mu \nu}}{\epsilon^n}\right) \tag{7.10}.$$
The $K^{(n)}$ depend on $\mu$ only by their functional dependence on the renormalized metric $G$. By the standard assumption that the bare metric do not depend on the renormalization scale process $\mu$, he concludes that
\begin{equation} \begin{aligned} &\frac{d G_{\mu \nu}}{d \ln{\mu}} = \epsilon G_{\mu \nu} + \beta^G_{\mu \nu}, & \beta^G_{\mu \nu} = \left( 1 - G \frac{\partial}{\partial G}\right)K^{(1)}_{\mu \nu},\\ &\beta^G_{\mu \nu} \frac{\partial}{\partial G}K^{(n)}_{\mu \nu} = \left( 1 - G \frac{\partial}{\partial G}\right)K^{(n+1)}_{\mu \nu}. \end{aligned} \tag{7.11} \end{equation} Here, the $ G \frac{\partial}{\partial G}K^{(1)}_{\mu \nu},$ is an abreviation for functional derivative of $K^{(1)}$ acting on $G$. However, I have no idea how he derived $(7.11)$. What happened to the other $K^{(n)}$ in the first equation on $(7.11)$? Also, I think the functonal derivative should be acting on $\frac{d G}{d \ln{\mu}}$, not on $G$, as we are differentiating the hole $(7.10)$ w.r.t. $\ln{\mu}$. What am I doing wrong?
