classical mechanics - Can potential be velocity dependent?

In the lagrangian solution for the equation of motion, there's a seemingly out of place $$\frac{\mathrm{d} }{\mathrm{d} t}\frac{\partial V}{\partial \dot{q_j}}$$

term. Potential energy is usually a function of the set of $x_i$ or position only. If $x_i$ can all be rewritten as functions of only $q_i$ and $t$, and $q_i$ can be varied without having to change $\dot{q_i}$. Then we're left with this term being precisely $0$

So at least for conservative forces, this term should equal zero. But where do we find cases where it isn't? Magnetic forces? Is it frictional forces (what is potential for a frictional force anyway? And does Lagrange's equation even work for inelastic systems, considering energy is not conserved?)