Jacobian

Calculus · Formula · Not yet linked from a lesson

Jij=∂fi∂xjJ_{ij} = \frac{\partial f_i}{\partial x_j}

The full matrix of partial derivatives for a function with several outputs — one row per output, one column per input. Autodiff never builds it in full; it multiplies by it one vector at a time.

Symbols

fif_i
The i-th output of the function.
xjx_j
The j-th input.
JJ
The matrix collecting every such pairing.
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