Let $(M^n,g)$ be a complete Riemannian manifold with $|Rm| \le 1$. Can we find two positive constants $C$ and $\epsilon$, depending only on $n$, such that under the normal coordinates $(g_{ij})$ with respect to any point $p \in M$, we have
$$
|\partial_k g_{ij}(x)| \le C
$$
for any $|x| \le \epsilon$?