Derivation of lagrange equation in classical mechanics

Question

I'm currently working on classical mechanics and I am stuck in a part of the derivation of the lagrange equation with generalized coordinates. I just cant figure it out and don't know if it's just basic mathematics or a physics assumption. I hope someone can explain this step to me. I don't understand the transition from the terms involving the partial derivative to the kinetic energy term.

Answer 1

Applying the derivative (don't forger the inner derivative): $$\frac{\partial}{\partial\dot{q}_{j}}\left(\frac{1}{2}m_{i}v_{i}^{2}\right)=\frac{1}{2}m_{i}\frac{\partial}{\partial\dot{q}_{j}}v_{i}^{2}=m_{i}\boldsymbol{v}_{i}\frac{\partial\boldsymbol{v}_{i}}{\partial\dot{q}_{j}}$$