Network Function Virtualization (NFV) is a promising paradigm which can increase network flexibility by deploying newly network services into networks. Because of the flexibility, NFV is also considered as one of the building blocks for 5G and edge computing. However, the intrinsic dynamic features of NFV and the rigorous requirements proposed by 5G and edge computing, such as delay, pose significant challenges to make the optimal Virtual Network Function (VNF) chain placement. To address this problem, we first formulate the VNF chain placement problem as an integer linear programming problem with taking the dynamic characteristics in to consideration. Then we propose an efficient dynamic algorithm, called DynAmic vnF placemenT (DAFT), which provides a good solution for VNF chain placement. DAFT is based on primal–dual technique and combines with an efficient subroutine which is solved by reducing to the shortest path problem. The theoretic analysis shows DAFT is (1-1$slash$e)-competitive to offline optimal solution. We also propose a modified algorithm concerning algorithm implementation issues, which is called Feasible DAFT (FDAFT). Finally, we evaluate the proposed algorithms through extensive numerical simulations. The experiment results show that DAFT and FDAFT achieve average competitive ratios up to 98.42% and 96.74%, respectively and outperform the compared algorithms around 50%.