In recent years, the availability of computer-based methods has created a revival of interests in exploring algebraic methods in nonlinear context. This note proposes a new approach to algebraic nonlinear observer design. After giving the notion of algebraic observability, and based on a novel algorithm of exact differentiation, the formulation of the nonlinear observer is realized via the construction of a set of linear time-varying differentiators. An example of a chemical reaction is given to show the effectiveness of our approach.