Let $g:\mathbb{R}^2 →\mathbb{R}^3$ be such that $g(0)=(1,1,1)$ and $ [Dg(0)]= \left[ {\begin{array}{cc} 1 & 1 \ \\1 & 1 \\1& 1 \ \end{array} } \right] $
Find $D_\mathbf{0}(\|g\|)(\mathbf{h})$ and $[D_\mathbf{0}(\|g\|)]$, where $\|·\| = \sqrt{⟨·, ·⟩}$ is the standard norm on $\mathbb{R}^3$.
By the chain rule:
$$D_0(\|g\|) = D_{g(0)}(\|\cdot\|) \circ D_0(g)$$
Hence:
$$[D_0(\|g\|)] = [D_{(1,1,1)} (\|\cdot\|)][D_0(g)]$$
Now recall that $D_x (\|\cdot\| )= \frac1{\|x\|} \langle x, \cdot \rangle $. Hence, $D_{(1,1,1)}(\|\cdot\|) h= \frac1{\sqrt 3} (h_1 + h_2 + h_3)$, which shows that:
$$[D_{(1,1,1)}(\|\cdot\|)] = \frac1{\sqrt 3}(1,1,1)$$
Therefore,
$$[D_0(\|g\|)] = \frac1{\sqrt 3} (1,1,1) \begin{pmatrix} 1 & 1 \\ 1 &1 \\ 1 &1 \end{pmatrix} = \sqrt 3 (1,1)$$