In $\frac{dU}{dt} = \sum_i \frac{\partial U}{\partial x_i}\frac{\partial x_i}{\partial t} + \frac{\partial U}{\partial t}$, does the second term come from the chain rule?

Yes, this is the chain rule for multivariable calculus. If you have some function $f$ of multiple variables, $f(x,y,...)$, then $\frac{df}{dt}= \frac{\partial f}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial f}{\partial y}\ \frac{\partial y}{\partial t}+ ...$