This paper aims at designing a non-asymptotic fractional order differentiator for a class of fractional order linear systems to estimate the Riemann-Liouville fractional derivatives of the output in discrete noisy environment. The adopted method is a recent algebraic method originally introduced by Fliess and Sira-Ramirez. Firstly, the fractional derivative of the output of an arbitrary order is exactly given by a new algebraic formula in continuous noise free case without knowing the initial conditions of the considered system. Secondly, a digital fractional order differentiator is introduced in discrete noisy cases, which can provide robust estimations in finite-time. Then, some error analysis is given, where an error bound useful for the selection of the design parameter is provided. Finally, numerical examples illustrate the accuracy and the robustness of the proposed fractional order differentiator.