The design of an ideal differentiator is a difficult and a challenging task. In this paper we discuss the properties and the limitations of two different structures of linear differentiation systems. The first time-derivative observer is formulated as a high-gain observer where the observer gain is calculated through a Lyapunov-like dynamical equation. The second one is given in Brunovski form in which the output to be differentiated appears as a control input and the differentiation gain is calculated from the dual Lyapunov equation of the first differentiation observer. A discrete-time version of the second form is given. Finally, illustrative examples are presented to show their strengths and weaknesses. (C) 2003 Elsevier Ltd. All rights reserved.