Downscaling global climate model outputs to regional and local scales often encounters significant challenges, particularly in accurately representing local climate variations such as precipitation. These challenges stem from the inherent biases in coarse-resolution climate model data and the difficulty of capturing fine-scale climatic and hydrological processes. Traditional downscaling methods, while improving spatial resolution, often fail to adequately address these biases, leading to inaccuracies in hydrological models that rely on such downscaled inputs. To address this issue, we propose a novel integration of a trainable bias correction feature within a differentiable hydrological model. This approach leverages a neural network to dynamically correct biases in precipitation inputs before they are utilized in hydrological computations. The bias correction network is a feedforward neural network designed to learn and correct the discrepancies based on the gradients arising from the final loss function based on streamflow. The integration involves preprocessing the input precipitation data from multiple CMIP6 global climate models (GCMs) basin-averaged to the CAMELS catchment scale within the conterminous US. This experiment represents a significant advancement over traditional static parameter downscaling methods by incorporating a dynamic, data-driven approach to bias correction. The bias correction network is trained, enabling it to adaptively reduce errors in real-time model inputs. This method is not only expected to improve the precipitation simulations from ensembles of GCMs but also enhances the overall reliability of hydrological predictions under diverse climatic conditions. The results are anticipated to demonstrate the effectiveness of the bias correction network in mitigating input biases and improving hydrological model outputs, highlighting the potential for differentiable models to advance bias correction practices in climate and hydrological modeling.