The ambiguity function is fundamental to radar waveform design, characterizing range and Doppler resolution capabilities. However, its traditional formulation involves non-differentiable operations, preventing integration with gradient-based optimization methods and modern machine learning frameworks. This paper presents the first complete mathematical framework and computational implementation for differentiable radar ambiguity functions. Our approach addresses the fundamental technical challenges that have prevented the radar community from leveraging automatic differentiation: proper handling of complex-valued gradients using Wirtinger calculus, efficient computation through parallelized FFT operations, numerical stability throughout cascaded operations, and composability with arbitrary differentiable operations. We term this approach GRAF (Gradient-based Radar Ambiguity Functions), which reformulates the ambiguity function computation to maintain mathematical equivalence while enabling gradient flow through the entire pipeline. The resulting implementation provides a general-purpose differentiable ambiguity function compatible with modern automatic differentiation frameworks, enabling new research directions including neural network-based waveform generation with ambiguity constraints, end-to-end optimization of radar systems, and integration of classical radar theory with modern deep learning. We provide complete implementation details and demonstrate computational efficiency suitable for practical applications. This work establishes the mathematical and computational foundation for applying modern machine learning techniques to radar waveform design, bridging classical radar signal processing with automatic differentiation frameworks.