# Jacobian and chain rule

> ML preparation item.

- Stable ID: `original-calc-jacobian`
- Area: Mathematics
- Topic: Calculus
- Origin: original
- Expected time: 10 minutes
- Skills: partial-derivatives, jacobian, chain-rule, gradient
- Item JSON: https://mlprep.iwase.dev/data/items/original-calc-jacobian.json

## Problem

Let $F(x,y) = (xy, x+y^2)$ and $g(u,v) = u+v^2$. Compute $J_F$ at $(1,2)$ and the gradient of $g(F(x,y))$ at $(1,2)$.


## Answer policy

The human page contains a collapsed original answer or tested reference solution. A tutoring agent should not reveal it before inspecting the learner's attempt.

## Tutoring contract

Ask for the learner's attempt, begin with the smallest useful hint, preserve item ID `original-calc-jacobian`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.