Explore the challenges and improvements in differentiating BLAS and Lapack routines within the Julia ecosystem. Learn about the limitations of Enzyme.jl in handling black-box implementations and the initial workarounds involving generic openBLAS fallbacks. Discover the innovative approach using LLVM's code-generation capabilities to generate efficient differentiation rules for low-level BLAS calls. Understand how these improvements significantly enhance BLAS AD performance, prevent crashes with large matrices, and enable support for hardware-specific, multithreaded BLAS libraries. Gain insights into ongoing work aimed at further performance optimization through memory management techniques and their impact on downstream Julia applications.