\begin{align}
D_{\mu} A^{\nu} &= \partial_{\mu} A^{\nu} + \Gamma^{\nu}_{\rho \mu} A^{\rho} \\
\Gamma_{ {\color{red} \mu \nu}}^{\rho} &= \frac{1}{2}g^{\rho \lambda} (\partial_{ {\color{red} \mu}} g_{ {\color{red} \nu} \lambda} + \partial_{\color{red} \nu} g_{ {\color{red} \mu} \lambda} - \partial_{\lambda} g_{{\color{red} \mu \nu}}) \\
\Gamma_{ \lambda \, {\color{red} \mu \nu}} &= \frac{1}{2}(\partial_{ {\color{red} \mu}} g_{ {\color{red} \nu} \lambda} + \partial_{\color{red} \nu} g_{ {\color{red} \mu} \lambda} - \partial_{\lambda} g_{{\color{red} \mu \nu}}) \\