  In this paper, we investigate the one-dimensional derivative nonlinear
Schr\"odinger equations of the form $iu_t-u_{xx}+i\lambda\abs{u}^k u_x=0$ with
non-zero $\lambda\in \Real$ and any real number $k\gs 5$. We establish the
local well-posedness of the Cauchy problem with any initial data in $H^{1/2}$
by using the gauge transformation and the Littlewood-Paley decomposition.
