\frac{\partial}{\partial\dot{q}_j}\sum_i\left(\frac{\partial V}{\partial q_i}\frac{d q_i}{dt}+\frac{\partial V}{\partial\dot{q}_i}\frac{d\dot{q}_i}{dt}\right)+\frac{\partial}{\partial\dot{q}_j}\frac{\partial V}{\partial t}