We consider the Schrödinger equation with derivative nonlinear term on the line. We obtain results on local well posedness under the assumption of uniqueness of weak solutions and we obtain the orbital stability of standing waves via the abstract theory of Grillakis, Shatah, Strauss. Moreover, we consider the Schrödinger equation with nonlinear derivative term on [0, +∞) under Robin boundary condition at 0. Using a virial argument, we obtain the existence of blowing up solutions and using variational techniques, we obtain stability and instability by blow up results for standing waves. Contents 1. Introduction 1 Acknowledgement 5 2. On the singularly perturbed derivative Schrödinger equation on the line 5 2.1. The Cauchy problem 5 2.2. The orbital stability of standing waves 9 3. On the derivative Schrödinger equation on the half line with Robin boundary condition 13 3.1. The existence of a blowing-up solution 13 3.2. Stability and instability of standing waves 15 References 24