In proof of Proposition 2.2 of the book Introduction to Modern Finsler Geometry by Shen and Shen I have faced the following problem:
Proposition 2.2: If $\sigma(t)$ is a geodesic on a Finsler manifold $(M,F)$, then the length of its tangent vector is constant, i.e., $F(\sigma(t),\dot{\sigma}(t))$ is constant.
While doing the proof it says $$\frac{d}{dt}[F^2(\sigma(t),\dot{\sigma}(t))]=\frac{\partial g_{ij}}{\partial x^k}\dot{\sigma}^k\dot{\sigma}^i\dot{\sigma}^j+\color{blue}{2}\frac{\partial g_{ij}}{\partial y^k}\ddot{\sigma}^k\dot{\sigma}^i\dot{\sigma}^j+2g_{ij}\ddot{\sigma}^i\dot{\sigma}^j.$$
We have calculated to expression in the LHS explicitly and got the same expression as RHS except for the $2$ in the second term as follows: $$\frac{d}{dt}[F^2(\sigma(t),\dot{\sigma}(t))]=\frac{\partial g_{ij}}{\partial x^k}\dot{\sigma}^k\dot{\sigma}^i\dot{\sigma}^j+\frac{\partial g_{ij}}{\partial y^k}\ddot{\sigma}^k\dot{\sigma}^i\dot{\sigma}^j+2g_{ij}\ddot{\sigma}^i\dot{\sigma}^j.$$ Can anyone help me resolve this problem? Any help is highly appreciated.