We establish exact rates of strong uniform consistency for the multivariate Nadaraya-Watson kernel estimator of the regression function and its derivatives. As a special case, we treat the local linear estimator of the regression and the local polynomial smoothers of derivatives of the regression in the more convenient univariate setting. Our methods of proofs are based upon modern empirical process theory in the spirit of the results of Einmahl and Mason [4] and Deheuvels and Mason [2] relative to uniform deviations of nonparametric kernel estimators.Key words: nonparametric regression, derivative estimation, kernel estimation, local linear least squares kernel estimator, local polynomial fitting, strong uniform consistency, rate of convergence, uniform limit law of the logarithm.

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