To solve the problem of the deadline-constrained task scheduling in the cloud computing system, this paper proposes a deadline-constrained scheduling algorithm for cloud computing based on the driver of dynamic essential path (Deadline-DDEP). According to the changes of the dynamic essential path of each task node in the scheduling process, the dynamic sub-deadline strategy is proposed. The strategy assigns different sub-deadline values to every task node to meet the constraint relations among task nodes and the user's defined deadline. The strategy fully considers the dynamic sub-deadline affected by the dynamic essential path of task node in the scheduling process. The paper proposed the quality assessment of optimization cost strategy to solve the problem of selecting server for each task node. Based on the sub-deadline urgency and the relative execution cost in the scheduling process, the strategy selects the server that not only meets the sub-deadline but also obtains much lower execution cost. In this way, the proposed algorithm will make the task graph complete within its deadline, and minimize its total execution cost. Finally, we demonstrate the proposed algorithm via the simulation experiments using Matlab tools. The experimental results show that, the proposed algorithm produces remarkable performance improvement rate on the total execution cost that ranges between 10.3% and 30.8% under meeting the deadline constraint. In view of the experimental results, the proposed algorithm provides better-quality scheduling solution that is suitable for scientific application task execution in the cloud computing environment than IC-PCP, DCCP and CD-PCP.