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With applications ranging from metabolomics to histopathology, quantitative phase microscopy (QPM) is a powerful label-free imaging modality. Despite significant advances in fast multiplexed imaging sensors and deep-learning-based inverse solvers, the throughput of QPM is currently limited by the speed of electronic hardware. Complementarily, to improve throughput further, here we propose to acquire images in a compressed form such that more information can be transferred beyond the existing electronic hardware bottleneck. To this end, we present a learnable optical compression-decompression framework that learns content-specific features. The proposed differentiable quantitative phase microscopy ($\partial μ$) first uses learnable optical feature extractors as image compressors. The intensity representation produced by these networks is then captured by the imaging sensor. Finally, a reconstruction network running on electronic hardware decompresses the QPM images. In numerical experiments, the proposed system achieves compression of $\times$ 64 while maintaining the SSIM of $\sim 0.90$ and PSNR of $\sim 30$ dB on cells. The results demonstrated by our experiments open up a new pathway for achieving end-to-end optimized (i.e., optics and electronic) compact QPM systems that may provide unprecedented throughput improvements.