We study the local convergence of a method presented by Cordero et al. of convergence order at least five to approximate a locally unique solution of a nonlinear equation. These studies show the convergence under hypotheses on the third derivative or even higher. The convergence in this study is shown under hypotheses on the first derivative. Hence, the applicability of the method is expanded. The dynamical analysis of this method is also studied. Finally, numerical examples are also provided to show that our results apply to solve equations in cases where earlier studies cannot apply. © 2015, Sociedad Española de Matemática Aplicada.