A disturbance estimator based on the design of an extended state observer (ESO) often considers that the time-derivative of the perturbation (or higher order time-derivatives) is very small and can be taken equal to zero in the observer design (standard extended system). In this paper, a better approximation of the disturbance dynamics is proposed using a backward difference method. A new extended system is designed based on this approximation. Any observer that makes the error dynamics exponentially stable for the standard extended system can then be used to estimate the state of the new extended system. The efficiency of the method is illustrated through an example.