dF = - p_inside * dV1 - p_outside * dV2 + σdS, where S is the bubble surface area.

This can then be used to find the correct equation for the difference in pressure. But intuitively I don't understand why. To find the pressure one must balance the different forces and is the above equation doing that? Also why is the last term positive? I.e. the negative signs come from the fact that p_inside is directed in the opposite direction as dV1 and similarly for the second term. When the surface expands does it then do positive work on the system?