The focus of this paper is the approximation of derivative information for a smooth function on a given triangulation. The paper extends a previous approach proposed by the same author to higher-order derivatives and to spaces with dimension two and higher. Given that the triangulation has certain properties, the author shows that the presented approach to approximate the derivatives achieves a higher accuracy than the already known methods. For this purpose, an appropriate averaging of available values is used. Numerical tests illustrate the presented result.