If the coordinate curves are geodesics, then $\partial_v E=\partial_u G=0$

Question

We have a parametrized regular surface and its coordinate curves in some parametrization are geodesics. I started by noting that, if $g_u, g_v$ denote the tangent vectors to those curves, then $\nabla_{g_i} g_i=0\implies \Gamma_{ii}^k=0$, that is, them being geodesics means the non null Christoffel symbols only have mixed indices downstairs. But this got me nowhere seeing as (denoting the components of the 1st fundamental form by E,F,G) $\partial_v E=\partial_v(g_u\cdot g_u)=2\Gamma_{uv}^k g_k\cdot g_u$. The other derivative is analogous. How do I proceed to prove $\partial_v E=\partial_u G=0$?