$D$-resultant for rational functions

In this paper we introduce the $D$-resultant of two rational functions $f(t),g(t) \in \mathbb{K}(t)$ and show how it can be used to decide if $\mathbb{K}(f(t),g(t))=\mathbb{K}(t)$ or if $\mathbb{K}[t]\subset \mathbb{K}[f(t),g(t)]$ and to find the singularities of the parametric algebraic curve define by $X=f(t), Y=g(t)$. In the course of our work we extend a result about implicitization of polynomial parametric curves to the rational case, which has its own interest.