Differential Matrix at the origin

Question

Assume function h : R → R s.t. $h(0) = (1,2)$ and $Dh(0) = \begin{bmatrix} 1& 2 &3 \\ 0 &0 &1 \end{bmatrix} $. Let $g:R^2 →R^2$ be the function of $g(x,y)=(x+2y+1,3xy)$. Find $D(g◦h)(0)$.

I am assuming D here is the differentiable jacobian matrix, but withouth knowing h, how would i possibly know how to compute the differential matrix of g◦h and then plug in 0?

Answer 1

Because chain rule! :) You know $Dh(0)$ and $D(g\circ h)(0) = Dg(h(0))Dh(0)$.