Let $M$ be a Riemannian manifold and $f$ a smooth function on $M$. The Bochner formula proved in Schoen-Yau's book "lectures in Differential Geometry": (prop. 2.2) $$ \Delta |\nabla f|^2(p)=2\sum _{ij} |f_{ij}|^2 +2 R_{ij}f_i f_j +2 \sum f_i(\Delta f)_i $$ The proof assumes that ${x_i}$ are normal coordinates around $p$. and $f_i$ is the co variant differential of $f$ with respect to $\partial/\partial x_i$. $R_{ij}$ is the Ricci tensor.
Then the proof proceed as follows: Since $|\nabla f|^2=\sum f_i^2$. Hence at $p$ one has $$ \Delta |\nabla f|^2 =\sum_j(\sum f_i^2)_{jj}=..... $$
My question is actually the first claim $$|\nabla f|^2=\sum f_i^2$$ It seems for my that in general $$\nabla f=f_sg^{si}\frac{\partial}{\partial x_i}$$ In normal coordinate we can only get the expression $|\nabla f|^2=\sum f_i^2$ only holds at the point $p$, not other points. But clearly we need to take derivative, the value at the point $p$ is not enough.
It's most likely I miss some point here, but I can't figure out why. Anyone can help?