In this paper, we prove some results on the existence and space-time decay properties of higher order derivatives in time and space variables of global strong solutions of the Cauchy problem for the Navier-Stokes equations in weighed L^\infty(\mathbb R^d,|x|^\gamma{\rm dx})\cap L^\infty(\mathbb R^d,|x|^\beta{\rm dx}) spaces. The estimate for the decay rate is optimal in the sense that it coincides with the decay rate of a solution to the heat equation.