An algorithm for adaptation of the gains of higher-order sliding mode-based exact differentiators is developed for the case where the upper bound for the rho th derivative of the tracking error signal exists but it is unknown. Unlikely other publications in the literature, the developed adaptive algorithm based on monitoring functions guarantees giobaiand exact tracking when used in closed-loop output feedback. In the closed- loop scenario, a global-exact and finite-time estimate for the variables (e) over dot(t), (e) over dot(t), . . . , e((rho-1)) (t) is applied to construct the sliding surface of the proposed sliding mode controller. The class of uncertain systems of arbitrary relative degree (p >= 1) takes into account time-varying perturbations with unknown bounds and state-dependent nonlinearities satisfying a linear growth condition with any unknown rate. The norm of the unmeasured state is majorised by using a hybrid state-norm estimator. Numerical examples and an engineering application to wing rock control are presented in order to illustrate the properties and advantages of the novel adaptation approach for sliding mode control design.