Person search suffers from the conflicting objectives of commonness and uniqueness between the two sub-tasks of person detection and re-identification that make the end-to-end training of person search networks difficult. In this paper, we propose a trident network for person search that performs detection, re-identification, and part classification together. We also devise a novel end-to-end training method using the adaptive gradient weighting function that controls the flow of back-propagated gradients through the re-identification and part classification networks according to the quality of the person detection. The proposed method not only prevents the over-fitting but encourages to exploit fine-grained features by incorporating the part classification branch into the person search framework. Experimental results demonstrate that the proposed method achieves the best performance among the state-of-the-art end-to-end person search methods.