New upper bounds are found in the problem of approximation of kth derivatives of a function of d variables defined on a simplex by the derivatives of an algebraic polynomial of degree at most n (0⩽k⩽n) interpolating the values of the function at equidistant nodes of the simplex. The estimates are obtained in terms of the diameter of the simplex, the angular characteristic introduced in the paper, the dimension d, the degree n of the polynomial, and the order k of the derivative to be estimated and do not contain unknown parameters. These estimates are compared with those most frequently used in the literature.