I'm trying to understand Takhtajan's "QM for Mathematicians" and I'm struggling still with the generalized coordinates. To make things simple consider a free particle on some Riemannian manifold. Takhtajan says on p.12 that, given generalized coordinates $(x_1,\ldots,x_n,v_1,\ldots,v_n)$ and a path $\gamma(t)=(\gamma_1(t),\ldots,\gamma_n(t),\gamma'_1(t),\ldots,\gamma'_n(t))$ in these coordinates, its Lagrangian is $$ L(\gamma(t),\gamma'(t)) = \frac{1}{2}g_{\gamma(t)}(\gamma'(t),\gamma'(t))=\frac{1}{2}\sum_{i,j=1}^n g\left(\frac{\partial}{\partial x_j},\frac{\partial}{\partial x_i}\right) \gamma'_i(t)\gamma'_j(t), $$ where $g$ is the Riemannian metric.
Now, I want to derive the Euler-Lagrange equations. How can I compute the terms $\frac{\partial L}{\partial v_i}$ or, on even lower level, $\frac{\partial \gamma'_j(t)}{\partial v_i}$?