Stating physical laws in terms of velocity

$$\frac{d}{dt}\left(\frac{\partial\mathcal{L}}{\partial v}\right)=\frac{\partial\mathcal{L}}{\partial x}=\frac{\partial\mathcal{L}}{\partial v}\frac{\partial v}{\partial x}$$

$$\frac{\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial v}}{\frac{\partial\mathcal{L}}{\partial v}}=\frac{\partial v}{\partial x}$$