We consider the initial value problem for cubic derivative nonlinear Schrödinger equations possessing weakly dissipative structure in one space dimension. We show that the small data solution decays like $O((\log t)^{-1/4})$ in $L^2$ as $t \to + \infty$. Furthermore, we find that this $L^2$-decay rate is optimal by giving a lower estimate of the same order. This is a joint work with Chunhua Li (Yanbian University), Yoshinori Nishii (Tokyo University of Science) and Hideaki Sunagawa (Osaka Metropolitan University).