We first present a local convergence analysis for some families of fourth and six order methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Earlier studies have used hypotheses on the fourth Frechet-derivative of the operator involved. We use hypotheses only on the first Frechet-derivative in one local convergence analysis. This way, the applicability of these methods is extended. Moreover, the radius of convergence and computable error bounds on the distances involved are also given in this study based on Lipschitz constants. Numerical examples illustrating the theoretical results are also presented in this study.