Apart from the vertex expansion discussed in Chap. 8 the other main strategy to obtain approximate solutions of the FRG flow equations is the derivative expansion. This method has been successfully applied to problems in statistical physics and field theory (see Bagnuls and Bervillier 2001, Berges et al. 2002, and Pawlowski 2007 for comprehensive reviews) where one is usually only interested in long-wavelength phenomena and an expansion of $$\varGamma_\varLambda[\bar{\varPhi}]$$ in gradients of the fields $$\bar{\varPhi}$$ seems reasonable. Such an expansion is justified by noting that, although the true generating functional of the irreducible vertices $$\varGamma[\bar{\varPhi}] = \varGamma_{\varLambda=0}[\bar{\varPhi}]$$ can contain nonanalyticities, the flowing $$\varGamma_\varLambda[\bar{\varPhi}]$$ is analytic for any finite value of the cutoff Λ.