Calculation of covariant derivative

Question

In the book "The Ricci flow:techniques and applications: Geometric Aspects" by B. Chow at page 9, there is an equation $$\Delta(\nabla_if)=\nabla_i(\Delta f)+R_{ik}g^{kl}\nabla_lf.$$ The author said the above equation can be derived from commuting covariant derivatives. But I can not get this equation. Please help me to find the intermediate steps

Answer 1

Commuting covariant derivatives implies
$$\nabla_i\nabla_j(\nabla^kf)=\nabla_j\nabla_i (\nabla^kf)+R^k_{ijl}\nabla^lf$$
Tracing over $k$ and $j$ yields $$\nabla_i\nabla_j(\nabla^jf)=\nabla_j\nabla_i (\nabla^jf)+R_{il}\nabla^lf$$
This will lead to your result.