ML Prep

Jacobian and chain rule

Problem

Let F(x,y)=(xy,x+y2)F(x,y) = (xy, x+y^2) and g(u,v)=u+v2g(u,v) = u+v^2. Compute JFJ_F at (1,2)(1,2) and the gradient of g(F(x,y))g(F(x,y)) at (1,2)(1,2).

Reveal answer or reference solution
JF(1,2)=[2114].J_F(1,2) = \begin{bmatrix} 2 & 1 \\ 1 & 4 \end{bmatrix}.

Since F(1,2)=(2,5)F(1,2)=(2,5), we have ∇g=(1,10)\nabla g=(1,10) in (u,v)(u,v) coordinates. Therefore,

JF(1,2)T∇g(F(1,2))=(12,41).J_F(1,2)^{\mathsf T}\nabla g(F(1,2)) = (12,41).

Local history

Loading attempts saved in this browser…

    Use with your agent

    Share this URL and your attempt. Ask the agent to start with a clarifying question or the smallest useful hint.

    Tutor me on https://mlprep.iwase.dev/item/original-calc-jacobian/. If window.mlPrepAgent is available, read attempts for item original-calc-jacobian before tutoring. Inspect my attempt, keep the item ID, and do not reveal the full answer first. After a real attempt, append its record with recorded_by agent, include agent_session_id when available, and read it back.

    Appears in