We present a detailed derivation of the one-particle-irreducible (1PI) differential renormalization-group generators originally developed by Nicoll and Chang and by Chang, Nicoll, and Young. We illustrate the machinery of the irreducible formulation by calculating to order ${\ensuremath{\epsilon}}^{2}$ the characteristic time exponent
z for the time-dependent Ginsburg-Landau model in the cases of conserved and nonconserved order parameter. We then calculate both
z and $\ensuremath{\eta}$ to order ${\ensuremath{\epsilon}}^{2}$ by applying to the 1PI generator an extension of the operator expansion technique developed by Wegner for the Wilson smooth-cutoff renormalization-group generator.
The Kadanoff theory of scaling near the critical point for an Ising ferromagnet is cast in differential form. The resulting differential equations are an example of the differential equations of the renormalization group. It is shown that the Widom-Kadanoff scaling laws arise naturally from these differential equations if the coefficients in the equations are analytic at the critical point. A generalization of the Kadanoff scaling picture involving an "irrelevant" variable is considered; in this case the scaling laws result from the renormalization-group equations only if the solution of the equations goes asymptotically to a fixed point.