Metric connection on a vector bundle at a point where the Riemannian metric of the base is standard to second order

Question

Let $E\to M$ be a smooth complex vector bundle with a hermitian metric, over a Riemannian manifold $M$, and let $\nabla$ be a metric connection on $E$. Suppose at a point $p\in M$, there is a coordinate neighborhood $(U,x_1,\dots,x_n)$ centered at $p$ such that the Riemannian metric of $M$ is standard to second order at $p$ (so that $g_{ij}(p)=\delta_{ij}$ and and the first-order partial derivatives of $g_{ij}$ vanishes at $p$). Then why is it true that $\nabla_{\partial_i}=\partial_i$ at $p$? (This is asserted in p.46 of Morgan's book on Seiberg-Witten Theory but I can't see why).