This paper aims at designing a fractional order differentiator based on a integer order linear system with zero initial conditions, where the fractional derivatives of the output are estimated using the output observation corrupted by a non zero-mean noise. Firstly, an integral algebraic formula for the fractional derivatives of the output is exactly obtained in continuous noise free case, using an appropriated modulating function. Unlike the improper integrals in the definitions of the fractional derivatives, the obtained formula is given by a proper integral. Then, an additional condition is added to the used modulating function in order to deal with the non zero-mean noise. After constructing the needed modulating function, a digital fractional order differentiator is proposed in discrete noisy case with some error analysis. Finally, the proposed fractional order differentiator is applied to design a fractional order PID controller for an integer order linear system.