This paper addresses the design in the time domain of fractional order differentiators for a class of signals satisfying a linear differential equation with unknown parameters. We first estimate the unknown parameters using modulating function method. Then, by applying generalized modulating function method and fractional integration by parts formula the left-sided Riemann-Liouville fractional derivative with an arbitrary order of a considered signal is given by two different integral formulae respectively. Thus, we take these integral formulae as proposed fractional order differentiators which do not contain any sources of errors in continuous noise free case. Unlike the improper integral in the definition of the left-sided Riemann-Liouville fractional derivative, the integrals in the proposed formulae can be proper and be considered as a low-pass filter by choosing appropriate modulating functions. Hence, digital fractional order differentiators applicable for on-line applications are deduced using a numerical integration method in discrete noisy case. Moreover, some error analysis are given for noise error contributions due to a class of stochastic processes. Finally, numerical examples are given to show the accuracy and robustness of the proposed fractional order differentiators.