ML Prep

Vector chain rule

Problem

Let F(x,y)=(x2+y,xy)F(x,y)=(x^2+y,xy) and h(u,v)=u2+3vh(u,v)=u^2+3v. Compute JF(1,2)J_F(1,2) and ∇(h∘F)(1,2)\nabla(h\circ F)(1,2).

Reveal answer or reference solution
JF(x,y)=[2x1yx],JF(1,2)=[2121]J_F(x,y)=\begin{bmatrix}2x&1\\y&x\end{bmatrix},\qquad J_F(1,2)=\begin{bmatrix}2&1\\2&1\end{bmatrix}

Now F(1,2)=(3,2)F(1,2)=(3,2) and ∇h(3,2)=(6,3)\nabla h(3,2)=(6,3). The chain rule gives

∇(h∘F)(1,2)=JF(1,2)T(6,3)T=(18,9)T\nabla(h\circ F)(1,2)=J_F(1,2)^{\mathsf T}(6,3)^{\mathsf T} =(18,9)^{\mathsf T}
Outcome

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