This paper proposes arbitrary-order distributed finite-time differentiator (AODFD) for leader-follower multi-agent systems (MAS) under directed graph by only using relative or absolute output information. By using arbitrary-order distributed finite-time differentiator via relative output information (AODFD-R), each follower agent can obtain the relative output information between itself and leader and the relative output's arbitrary-order derivatives, where the information to be measured is only the local relative output information between each follower agent and its neighboring agents. As a simple extension of AODFD-R, the arbitrary-order distributed finite-time differentiator via absolute output information (AODFD-A) is also given. The finite-time stability of the closed-loop system under AODFD is proved by constructing a Lyapunov function skillfully. Finally, several simulation examples are given to verify the effectiveness of the AODFD.