# Munkres Analysis on Manifolds - Differentiation

Let $f: \mathbb{R}^3 \to \mathbb{R}^2$ satisfy the conditions $f(0)=(1,2)$ and $$Df(0)=\left[ \begin{array}{ccc} 1 & 2 & 3 \\ 0 & 0 & 1 \\ \end{array} \right]$$ Let $g: \mathbb{R}^2 \to \mathbb{R}^2$ be defined by the equation $$g(x,y)=(x+2y+1,3xy).$$ Find $D(gof)(0)$.

• Hint: What is $Dg(1,2)$? Oct 12, 2012 at 15:18

Using the chain rule, you get $D(g\circ f)(0)={Dg}_{|_{f(0)}}\cdot Df(0)=Dg(1,2)\cdot Df(0)=\left[ \begin{array}{ccc} 1 & 2 \\ 3\cdot 2 & 3\cdot 1 \\ \end{array} \right]\cdot \left[ \begin{array}{ccc} 1 & 2 & 3 \\ 0 & 0 & 1 \\ \end{array} \right]=\left[ \begin{array}{ccc} 1 & 2 & 5 \\ 6 & 12 & 21 \\ \end{array} \right]$