This problem comes from the book Hamilton's Ricci flow.

Given a smooth functional $f$, and following system. $$\partial_tg_{ij}=-2(R_{ij}+\nabla_i\nabla_jf)$$ If there exist a 1 parameter family of diffeomorphism $\Psi(t):M\to M$ by $$\partial_t\Psi(t)=\nabla_{g(t)}f(t), \Psi(0)=id_M$$ Show that $\tilde{g}_{ij}:=\Psi(t)^*g(t)$ satisfy $$\partial_t \tilde{g}_{ij}=-2\tilde{R}_{ij}$$

My progress is \begin{align*}\partial_t\tilde{g}_{ij}&=\partial_{t}g(\Psi(t)_*(\partial_i),\Psi(t)_*(\partial_j))\\ &=(\partial_tg)(\Psi(t)_*(\partial_i),\Psi(t)_*(\partial_j))+g(\partial_t\Psi(t)_*(\partial_i),\Psi(t)_*(\partial_j))+g(\Psi(t)_*(\partial_i),\partial_t\Psi(t)_*(\partial_j)) \\ &=-2(\tilde{R}_{ij}+\tilde{\nabla}_i\tilde{\nabla}_jf)+g(\partial_t\Psi(t)_*(\partial_i),\Psi(t)_*(\partial_j))+g(\Psi(t)_*(\partial_i),\partial_t\Psi(t)_*(\partial_j)) \end{align*} So my question is how to show $$g(\partial_t\Psi(t)_*(\partial_i),\Psi(t)_*(\partial_j))+g(\Psi(t)_*(\partial_i),\partial_t\Psi(t)_*(\partial_j)) =2 \tilde{\nabla}_i\tilde{\nabla}_jf$$