We study the local convergence of a Newton-like method of convergence order six to approximate a locally unique solution of a nonlinear equation. Earlier studies show convergence under hypotheses on the seventh derivative or even higher. The convergence in this study is shown under hypotheses on the first derivative although only the first derivative appears in these methods. Hence, the applicability of the method is expanded. Finally, we solve the problem of the fractional conversion in the ammonia process showing the applicability of the theoretical results.