For the random design nonparametric regression, we use the double smoothing technique to construct kernel estimators of the regression function and its derivatives. As an estimator of the regression function, this double smoothing estimator (DSE) shares the superior asymptotic variance quantity of the Nadaraya-Watson estimator (NWE) and the nice asymptotic bias quality of the Gasser-Mueller estimator (GME). Based on the asymptotic mean square error (AMSE), the DSE is uniformly better than the GME. To estimate derivatives of the regression function, the estimators derived by the DSE give insight directly and good interpretability, and are also uniformly better than those derived by the GME, in terms of the AMSE. Under the regularity conditions, these DSE of the regression function and its derivatives are asymptotically normal.