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).

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