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Derivatives, Gradients, Jacobians and Hessians – Oh My!

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demofox

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Math General

Description

Derivatives, Gradients, Jacobians and Hessians - Oh My is a single educational blog post by demofox on The blog at the bottom of the sea that explains how these four calculus tools relate.

It starts with scalar derivatives for finding minima, then gradients as vectors of partial derivatives for higher-dimensional descent and SDF antialiasing. It builds Jacobian matrices for multivariable maps that warp space, including determinants for scale and invertibility, and Hessian matrices of second derivatives for curvature-aware optimization and costly ML cases with quasi-Newton alternatives.

It is for intermediate readers who know basic calculus and want graphics- and ML-oriented intuition rather than a formal textbook. Shader or optimization work is helpful context but not required.

Access is a free HTML post with equations and prose walkthroughs; no download or login is needed.

It is useful as a compact bridge from matrix names to concrete uses in optimization and rendering space warps.

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