In this paper, new iterative method is presented of fifth-order for solving non-linear equations f (x) = 0 a devoid of the second derivative which requires two derivative functions and evaluations for each step, using both weight functions and synthesis techniques together. This method improves Newton's method and thus the efficiency index has been improved from 1.414 to 1.495. The convergence analysis for the new method is discussed. We provide some numerical examples that illustrate the performance of our proposed method by comparing them with numerical methods of fifth-order also the complex dynamics and basins of attraction is discussed, comparing it with several methods of the same order, thus comparisons show that new method gives the best results.