When traversing a symmetry-breaking second-order phase transition at a finite rate, topological defects form whose number dependence on the quench rate is given by simple power laws. We propose a general approach for the derivation of such scaling laws that is based on the analytical transformation of the associated equations of motion to a universal form rather than employing plausible physical arguments. We demonstrate the power of this approach by deriving the scaling of the number of topological defects in both homogeneous and nonhomogeneous settings. The general nature and extensions of this approach are discussed.