We address the problem of approximating the derivatives of a differentiable function of variables () by the derivatives of a polynomial on an -simplex for the standard method of interpolation by Lagrange polynomials at the points of a uniform grid on this simplex. For the error of approximation of these derivatives by the derivatives of the interpolation polynomial, we obtain upper bounds expressed in terms of new geometric characteristics of the simplex. The proposed characteristics of the simplex are clear and easy to calculate.