The derivation of the Christoffel symbol can be achieved by analyzing the vanishing of the covariant derivative of the metric tensor. The equation g_{\rho\sigma,\mu}=\Gamma^\lambda_{\mu\rho}g_{\lambda\sigma}+\Gamma^\lambda_{\mu\sigma}g_{\rho\lambda} is crucial in this derivation. Additionally, the expression g_{\mu\sigma,\rho}+g_{\mu\rho,\sigma}-g_{\rho\sigma,\mu} aids in simplifying the calculations. It is essential to remember that the Levi-Civita connection is torsion-free, which ensures the symmetry of the Christoffel symbol in its lower indices.
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