Christoffel symbol at local inertial frame

Question

As I can always express an local point on a manifold to be local inertial frame, the Christoffel symbol is always zero at that particular point. In Carrol's Spacetime and Geometry, Eq (3.128) shows the derivation of Riemann tensor in terms of metric at a local inertial frame where the Christoffel symbol vanishes. However, why its derivatives do not vanish? Derivative of zero is still zero.

Answer 1

The existence of Riemann normal coordinates means one can find coordinates at a point $p$ such that the metric can be written as $$ g_{\mu \nu}(p) = \eta_{\mu \nu} \quad , \quad g_{\mu \nu , \lambda}(p)= 0 \ , $$ so we also have $\Gamma^{\lambda}_{\mu \nu}(p) = 0$. This does not mean that $g_{\mu \nu , \lambda \sigma}(p)= 0$, so in general $\partial_\rho \Gamma^{\lambda}_{\mu \nu}(p)\neq 0$.

As the other comments say, a function can be $0$ at a point but have non-zero derivative; similarly, its derivative vanishing at a point doesn't mean its second derivative is nonzero, etc.

See section 3.3.1 of these lecture notes for details of how the existence of normal coordinates can be shown or for more details in general.