The negative gradient flow of Einstein-Hilbert functional in a fixed conformal class is Yamabe flow

Question

In the book "Hamilton's Ricci flow" written by Bonnett Chow, Peng Lu and Lei Ni, there is an exercise pullzed me: Show that $n \ge 3$, the negative gradient flow of $$E(g)=\int_{\mathcal{M}} R d \mu$$ in a fixed conformal class (i.e. in $[g_{0}] = e^{u}g_{0}$ for some $u \in C^{\infty}(M)$) is the Yamabe flow $$\frac{\partial}{\partial t} g = -R g$$ How can I prove it? Is there any solutions or references?