  This paper is concerned with the derivative nonlinear Schrodinger equation
with periodic boundary conditions
  $$\mathbf{i}u_t+u_{xx}+\mathbf{i}\Big(f(x,u,\bar{u})\Big)_x=0,\quad
x\in\mathbb{T}:=\mathbb{R}/2\pi\mathbb{Z},$$ where $f$ is an analytic function
of the form $$f(x,u,\bar{u})=\mu|u|^2u+f_{\geq4}(x,u,\bar{u}),\quad
0\neq\mu\in\mathbb{R},$$ and $f_{\geq4}(x,u,\bar{u})$ denotes terms of order at
least four in $u,\bar{u}$. We show the above equation possesses Cantor families
of smooth quasi-periodic solutions of small amplitude. The proof is based on an
infinite dimensional KAM theorem for unbounded perturbation vector fields.
