Confusion about the Generalized Expression of Force from the Lagrangian

The force in generalized coordinate system $${x}^{\alpha}=({x}^{1},{x}^{2},{x}^{3})$$ is defined as \begin{align*} F^{\omega}\stackrel{\mathrm{def}}=g^{\omega\alpha}\bigg(\frac{\mathrm{d}}{\mathrm{d}\lambda}\frac{\partial{T}}{\partial{{v}^{\alpha}}}-\frac{\partial{T}}{\partial{{x}^{\alpha}}}\bigg) \end{align*} Where $${v}^{\alpha}=\frac{\mathrm{d}{x}^{\alpha}}{\mathrm{d}\lambda}$$ is the velocity and $$T=\frac{m}{2}g_{\mu\nu}{v}^{\mu}{v}^{\nu}$$ is the kinetic energy. Im aware that for calculating the derivatives i should arrive at this expression in the end \begin{align*} F^{\omega}=m\bigg(\frac{\mathrm{d}{v}^{\omega}}{\mathrm{d}\lambda}+\Gamma^{\omega}_{\mu\nu}{v}^{\mu}{v}^{\nu}\bigg) \end{align*} However when partially deriving the kinetic energy with respect to the spacial coordinates i just get this \begin{align*} \frac{\partial{T}}{\partial{x}^{\alpha}}=\frac{m}{2}{v}^{\mu}{v}^{\nu}\frac{\partial{g_{\mu\nu}}}{\partial{x}^{\alpha}}=\frac{m}{2}(\Gamma^{\beta}_{\mu\alpha}g_{\nu\beta}+\Gamma^{\beta}_{\alpha\nu}g_{\mu\beta}){v}^{\mu}{v}^{\nu}={m}\Gamma^{\beta}_{\mu\alpha}g_{\nu\beta}{v}^{\mu}{v}^{\nu} \end{align*} Which is wrong since $$\Gamma^{\beta}_{\mu\alpha}g_{\nu\beta}g^{\omega\alpha}\neq-\Gamma^{\omega}_{\mu\nu}$$ or is it? Can someone please explain to me where i have made a mistake?

\begin{align*} \frac{\partial{T}}{\partial{v}^{\alpha}}=\frac{m}{2}g_{\mu\nu}({v}^{\mu}\delta^{\nu}_{\alpha}+{v}^{\nu}\delta^{\mu}_{\alpha})={m}g_{\mu\alpha}{v}^{\mu} \end{align*} \begin{align*} \frac{\mathrm{d}}{\mathrm{d}\lambda}\frac{\partial{T}}{\partial{v}^{\alpha}}={m}\frac{\mathrm{d}(g_{\mu\alpha}v^{\mu})}{\mathrm{d}\lambda}={m}g_{\mu\alpha}\frac{\mathrm{d}{{v}^{\mu}}}{\mathrm{d}\lambda}+{m}\frac{\partial{g_{\mu\alpha}}}{\partial{x^{\beta}}}{v}^{\beta}{v}^{\mu}={m}g_{\mu\alpha}\frac{\mathrm{d}{{v}^{\mu}}}{\mathrm{d}\lambda}+{m}(\Gamma^{\gamma}_{\mu\beta}g_{\gamma\alpha}+\Gamma^{\gamma}_{\alpha\beta}){v}^{\beta}{v}^{\mu} \end{align*} Now putting all terms together like in the definition gives the result \begin{align*} \frac{\mathrm{d}}{\mathrm{d}\lambda}\frac{\partial{T}}{\partial{v}^{\alpha}}-\frac{\partial{T}}{\partial{x}^{\alpha}}={m}g_{\gamma\alpha}\bigg(\frac{\mathrm{d}{{v}^{\gamma}}}{\mathrm{d}\lambda}+\Gamma^{\gamma}_{\mu\beta}{v}^{\mu}{v}^{\beta}\bigg) \end{align*}