Based on the de Casteljau algorithm for triangular patches, also using some existing identities and elementary inequalities, this paper presents two kinds of new magnitude upper bounds on the lower derivatives of rational triangular Bézier surfaces. The first one, which is obtained by exploiting the diameter of the convex hull of the control net, is always stronger than the known one in case of the first derivative. For the second derivative, the first kind is an improvement on the existing one when the ratio of the maximum weight to the minimum weight is greater than 2; the second kind is characterized as being represented by the maximum distance of adjacent control points.