A new robust exact sliding mode (SM) based differentiator is proposed which provides for the fast global finite-time convergence of its outputs to the first n exact derivatives of its input f(t). The differentiator utilises the knowledge of a function L(t) providing the estimation vertical bar f(0)((n+1))vertical bar <= L(t), and satisfying vertical bar(L) over dot vertical bar/L <= M for a known bound M. The standard accuracy of the homogeneous SM-based differentiator is preserved in the presence of discrete sampling and noises in both f and L. The proposed discretisation scheme ensures the same accuracy in computer realisation.