When the nth derivative of a signal to be differentiated is bounded by a known constant, Levant's arbitrary-order differentiator provides, in the absence of noise, for an exact estimation of all derivatives up to order (n - 1) in finite time. Recently, Levant has shown that if the nth derivative is bounded by a known time-varying function, the same differentiator with a time-varying gain is also able to estimate in finite time all derivatives, if the initial condition of the differentiation error is sufficiently small. In this paper, we find, using a smooth Lyapunov function, under which conditions the same result is valid globally.