Invariant Relations for the Derivatives of Two Arbitrary Polynomials

Suppose $P(x)$ and $Q(x)$ are two arbitrary polynomials. In this paper we use the theory of resultants of two polynomials including Sylvester's Matrix to specify a large number of polynomial relations involving $P(x)$, $Q(x)$ and their derivatives. We also include a research problem for the reader to consider.