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.