  In this work we shall consider the initial value problem associated to the
generalized derivative Schr\"odinger equations \begin{equation*} \p_tu=i\p_x^2u
+ \mu\,|u|^{\a}\p_xu, \hskip10pt x,t\in\R, \hskip5pt 0<\a \le 1\;\, {\rm
and}\;\, |\mu|=1, \end{equation*} and \begin{equation*} \p_tu=i\p_x^2u +
\mu\,\p_x\big(|u|^{\a}u\big), \hskip10pt x,t\in\R, \hskip5pt 0<\a \le 1\;\,
{\rm and}\;\, |\mu|=1. \end{equation*} Following the argument introduced by
Cazenave and Naumkin \cite{Cazenave} we shall establish the local
well-posedness for a class of small data in an appropriate weighted Sobolev
space. The other main tools in the proof include the homogeneous and
inhomogeneous versions of the Kato smoothing effect for the linear
Schr\"odinger equation established by Kenig-Ponce-Vega in \cite{KPV1}.
