This paper deals with the problem of interval estimation. The proposed approach is based on the use of successive derivatives in continuous time (or forward measurements in discrete time) in an algebraic (or local) interval observer. The main advantage of such an observer is that it relaxes the Metzler assumption widely used in the literature of cooperative interval observers. Moreover, no prior knowledge on the initial condition is needed, and no stability nor convergence issue do appear as the estimation is local. After introducing such an observer for Linear Time Variant (LTV) systems with bounded disturbances, some extensions are proposed for Linear Parameter Varying (LPV) systems as well as for functional systems and systems affected by Unknown Input (UI). Some examples illustrate the theoretical contributions.