formula

$\lim_{t\to t_0}\frac{P_{t_0t}^{-1}V(t)-V(t_0)}{t-t_0}&=&\bigg(\lim_{t\to t_0}\frac{\tilde{V}^j(t)-\tilde{V}^j(t_0)}{t-t_0}\bigg)\partial_j(\gamma(t_0))\\ &=&\dot{\tilde{V}}^j(t_0)\partial_j(\gamma(t_0))$