The purpose of this note is to present the effects induced on the dynamics of a linear system using a derivative operator which is implemented by means of a delay-difference approximation. In this sense, a linear time-invariant (LTI) single-input/single-output (SISO) system is considered, subject to a PD-type controller for which the derivative action was implemented by considering the aforementioned mechanism. The use of such a methodology leads to a characteristic function with coefficients that explicitly depend on the delay parameter. This note presents sufficient conditions under which the approximation leads to unstable (real) roots, showing explicitly the sensitivity of the approximation with respect to "small" delays. Several illustrative examples complete the presentation.