The paper introduces the concept of *differentiable microstructures*, which are parameterized microstructures that exhibit continuous variations in both geometry and mechanical properties. The authors propose a novel formulation for topology optimization to construct such microstructures, starting with a low-volume fraction microstructure and generating a series of microstructures with increasing volume fractions. The optimization process maximizes the bulk modulus of the entire series, ensuring that the resulting microstructures are geometrically and physically differentiable. This approach eliminates the need for quantization in two-scale optimization and avoids the requirement for multi-material optimization, making it more efficient and flexible. The method is demonstrated through a series of experiments, including the optimization of 2D microstructures to maximize their bulk modulus. The results show that the proposed approach yields microstructures with bulk moduli that closely approach the theoretical limit, outperforming traditional uniform morphing methods and alternative approaches. The paper also discusses the geometric features and differentiability of the optimized outcomes, providing heuristic principles for generating input microstructures under specified conditions. Finally, the benefits of differentiable microstructures are validated in two-scale topology optimization, showing improved performance compared to decoupled two-scale optimization methods. The paper concludes with a discussion on future research directions, emphasizing the potential of differentiable microstructures in designing multi-scale structures with superior functionalities.The paper introduces the concept of *differentiable microstructures*, which are parameterized microstructures that exhibit continuous variations in both geometry and mechanical properties. The authors propose a novel formulation for topology optimization to construct such microstructures, starting with a low-volume fraction microstructure and generating a series of microstructures with increasing volume fractions. The optimization process maximizes the bulk modulus of the entire series, ensuring that the resulting microstructures are geometrically and physically differentiable. This approach eliminates the need for quantization in two-scale optimization and avoids the requirement for multi-material optimization, making it more efficient and flexible. The method is demonstrated through a series of experiments, including the optimization of 2D microstructures to maximize their bulk modulus. The results show that the proposed approach yields microstructures with bulk moduli that closely approach the theoretical limit, outperforming traditional uniform morphing methods and alternative approaches. The paper also discusses the geometric features and differentiability of the optimized outcomes, providing heuristic principles for generating input microstructures under specified conditions. Finally, the benefits of differentiable microstructures are validated in two-scale topology optimization, showing improved performance compared to decoupled two-scale optimization methods. The paper concludes with a discussion on future research directions, emphasizing the potential of differentiable microstructures in designing multi-scale structures with superior functionalities.