---
title: "The Jacobian Matrix"
description: "Build the Jacobian of a map from two inputs to two outputs: one row per output, one column per input, each entry a partial derivative you can test."
canonical_url: "https://fanout.sh/ai/lessons/math-fundamentals-jacobian-matrix"
md_url: "https://fanout.sh/ai/lessons/math-fundamentals-jacobian-matrix.md"
access: "pro-preview"
---

# The Jacobian Matrix

Build the Jacobian of a map from two inputs to two outputs: one row per output, one column per input, each entry a partial derivative you can test.

- Access: Fanout Pro (public preview)
- Module: Math Fundamentals

## A glimpse of The Jacobian Matrix

A function can turn two inputs into two outputs. The Jacobian is a table of local rates: each row tracks one output, and each column tracks one input.

For y₁ = 2x₁ + x₂, a small increase in x₁ changes y₁ twice as much. The first row of the Jacobian records the rates 2 and 1.

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