We study the derivative expansion for the effective action in the framework of the Exact Renormalization Group for a single component scalar theory. By truncating the expansion to the first two terms, the potential U_k and the kinetic coefficient Z_k, our analysis suggests that a set of coupled differential equations for these two functions can be established under certain smoothness conditions for the background field and that sharp and smooth cut-off give the same result. In addition we find

more »

... hat, differently from the case of the potential, a further expansion is needed to obtain the differential equation for Z_k, according to the relative weight between the kinetic and the potential terms. As a result, two different approximations to the Z_k equation are obtained. Finally a numerical analysis of the coupled equations for U_k and Z_k is performed at the non-gaussian fixed point in D<4 dimensions to determine the anomalous dimension of the field.