The existence and concentration of standing waves were studied for a class of nonlinear Schrodinger equations with first order derivative terms.Preci-sely,as the small parameter(which corresponds to Planck constant) approches to zero,the standing waves of these nonlinear Schrodinger equations concentrate at the non degenerate critical points of potentials.The paper concerns the singular perturbation of elliptic equations with first order derivative terms and this is also a new feature of the paper.