The discretization of fractional-order differentiator is the key step in digital implementation of a fractional-order controller. This paper focuses on the Tustin transform based direct discretization methods for fractional-order differentiator. Firstly, fractional order differintegral and its discretization are reviewed briefly. Secondly, power series expansion, continued fractional expansion and Muir-recursion expansion are presented and the algorithm expressions are introduced. The error function is defined and an illustrative example is given to compare the advantages and disadvantages of the discretization methods. The simulations and comparisons show that continued fractional expansion has better properties in a wide frequency band for fractional-order differentiator, while the computational complexity is high. Power series expansion and Muir-recursion expansion have similar properties and the former has the advantage of computational efficiency. It is necessary to select suitable discretization method in the implementation of digital fractional order controller.