{
  "schema": "ml-prep/item@1",
  "item": {
    "id": "original-calc-jacobian",
    "area": "mathematics",
    "topic": "calculus",
    "origin": "original",
    "title": "Jacobian and chain rule",
    "skills": [
      "partial-derivatives",
      "jacobian",
      "chain-rule",
      "gradient"
    ],
    "priority": "core",
    "difficulty": "medium",
    "estimated_minutes": 10,
    "prerequisites": [
      "partial-derivatives"
    ],
    "prompt": "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 $(1,2)$.\n",
    "answer": "$$\nJ_F(1,2) = \\begin{bmatrix} 2 & 1 \\\\ 1 & 4 \\end{bmatrix}.\n$$\nSince $F(1,2)=(2,5)$, we have $\\nabla g=(1,10)$ in $(u,v)$ coordinates. Therefore,\n$$\nJ_F(1,2)^{\\mathsf T}\\nabla g(F(1,2)) = (12,41).\n$$\n",
    "check": {
      "kind": "numeric",
      "id": "original-calc-jacobian",
      "values": [
        2,
        1,
        1,
        4,
        12,
        41
      ]
    }
  }
}