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The metric Bezout Theorem proved in an earlier paper can be extended to aderivative version that compares derivatives of the algebraic distance of apoint $\theta$ to two properly intersecting cycles in projective space with thederivatives of the algebraic distance of $\theta$ to their intersection.  This improvement can be used to make algebraic independence criteria, to beproved in a forthcoming paper, more flexible and to refine Approximationresults that are proved using the metric Bezout Theorem.

Diophantine Approximation on Varieties IV: Derivated algebraic distance and derivative metric Bezout Theorem