Compute $\frac{\partial}{\partial t}\mathcal{F}(g_{ij},f)$

Question

How to get the equality 1 ?

When I compute it ,I get stuck in $\frac{\partial}{\partial t}R_{ij}$ and $\frac{\partial}{\partial t}(\nabla_if\nabla_jf)$. I don't know how to deal the two terms.

Below picture is from 201th page of this paper.

Answer 1

Basically, just use the lemma 1.5.2 in that paper. Replace, knowing that the first variation of the metric is $v_{ij} = -2 \text{Ric}_{ij}$, so that $v=-2R$, and the first variation $h$ of the function $f$ is $\frac{\partial f}{\partial t}$, which is also $h = - \Delta f + |\nabla f|^2 -R$. Does this answer your question?