2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/217432The metric Bezout Theorem proved in an earlier paper can be extended to a derivative version that compares derivatives of the algebraic distance of a point $θ$ to two properly intersecting cycles in projective space with the derivatives of the algebraic distance of $θ$ to their intersection. This improvement can be used to make algebraic independence criteria, to be proved in a forthcoming paper, more flexible and to refine Approximation results that are proved using the metric Bezout Theorem.Algebraic GeometryNumber Theory14G40, 11G35, 11G50, 11J13, 14J20text