In this paper we will present two derivative free iterative methods for finding the root of a nonlinear equation f (x) = 0. The new methods are based on direct and inverse polynomial interpolation. We will prove that one of the methods has fifth-order convergence and the other method sixth-order convergence. Several examples will show that convergence and the efficiency of the new methods to be better than the classic Newton's method and others derivative free methods with high order of convergence previously presented.