Differentiation of resultants and common multiple roots of polynomials

In the article it is proven that once polynomials f and g possess a common root w of multiplicity s for f and multiplicity p for g, the Taylor series expansion for their resultant R = R (f , g)(g , b) in variables b (coefficients g) starts with the summand of order s, and the Taylor series expansion for R = R (f , g)(g , b) in variables a (coefficients f) starts with the summand of order p; and the explicit formulas for the corresponding summands of the Taylor series are obtained. Based on this, a number of results linking higher derivatives of resultants and multiple common roots of polynomials, which differ in ideas from the well-known ones, are obtained.