In this paper, we propose a global exact differentiator with dynamic gains based on higher-order sliding modes to solve the problem of global trajectory tracking via output-feedback for a class of uncertain nonlinear plants with disturbances. Norm observers for the unmeasured state are employed to dominate the disturbances as well as to adapt the gains of the proposed differentiator since the nonlinearities may be state-dependent and time-varying. Differently from some previous works in the literature, no hybrid switching scheme is necessary to state global stability using only input-output information. For the first time, uniform global exponential stability and ultimate exact tracking are guaranteed exclusively employing higher-order sliding modes based exact differentiators. Numerical simulations are presented to validate the analysis and show the effectiveness of the proposed method.