I think I was just confused by the notation, but if anybody was going to post in this thread, let me know if this is right:
The right-hand side is the two partial derivatives added together while the left-hand side is just a regular derivative? The intention is to set the demand function (left) equal to the supply function (right) to achieve equilibrium. Since the supply function is a function of two variables, the partials need to be taken. So the right-hand side is the sum of those partial derivatives? Is this correct?