One challenge that still exists in computer vision, particularly when motions are complex or when blur is spatially varied, is the task of blur deblurring in blind image restoration. Despite the advanced capabilities of deep learning-based methods, most current approaches either lack fine-grained edge representation or fail to aggregate global contextual information, which adversely affects t-he overall perceptual quality of the results. Traditional deblurring models often focus on either local-text recovery or whole-structure recovery, but seldom on both. These weaknesses inspire the idea of DGAI-Net (Deep Gradient-Aware Integration Network). This novel architecture is an interpretable neural network integrating classical mathematical operations, differentiation, and integration into the learning process as trainable modules within the learning pipeline. There are two main modules in our architecture: (1) a Gradient-Aware Derivative Module that improves edge preservation by calculating spatial derivatives with fixed or learnable Sobel-like operators, and (2) an Integration Module that uses directional cumulative feature aggregation that emulates integral behavior in reconstructing damaged structures. We test our approach on the REDS dataset, which uses synthesized blur, and compare our results to the latest approaches to the problem. The findings demonstrate that DGAI-Net outperforms previous methods in terms of quantitative evaluation (e.g., PSNR, SSIM) and visual sharpness. The network exhibits better reconstruction at the edges and textures, which demonstrates the significance of having a calculus-inspired prior embedded in the network structure. It is worth pursuing dynamic weight tradeoffs between gradient and integration cues, as well as generalizing DGAI-Net to actual blind deblurring applications.