Benefiting from the strong capabilities of deep CNNs for feature representation and nonlinear mapping, deep-learning-based methods have achieved excellent performance in single image super-resolution. However, most existing SR methods depend on the high capacity of networks that are initially designed for visual recognition, and rarely consider the initial intention of super-resolution for detail fidelity. To pursue this intention, there are two challenging issues that must be solved: (1) learning appropriate operators which is adaptive to the diverse characteristics of smoothes and details; (2) improving the ability of the model to preserve low-frequency smoothes and reconstruct high-frequency details. To solve these problems, we propose a purposeful and interpretable detail-fidelity attention network to progressively process these smoothes and details in a divide-and-conquer manner, which is a novel and specific prospect of image super-resolution for the purpose of improving detail fidelity. This proposed method updates the concept of blindly designing or using deep CNNs architectures for only feature representation in local receptive fields. In particular, we propose a Hessian filtering for interpretable high-profile feature representation for detail inference, along with a dilated encoder-decoder and a distribution alignment cell to improve the inferred Hessian features in a morphological manner and statistical manner respectively. Extensive experiments demonstrate that the proposed method achieves superior performance compared to the state-of-the-art methods both quantitatively and qualitatively. The code is available at github.com/YuanfeiHuang/DeFiAN.