\begin{align}
f(x,y,z) &= (u(x,y,z),v(x,y,z))\\
f(0,0,0) &= (1,2)\\
g(u,v) &= (u+2v+1,3uv)
\end{align}\
\begin{align}
Dg(u,v) &= \left( {\begin{array}{*{20}{c}}
{{\partial _u}(u + 2v + 1)}&{{\partial _v}(u + 2v + 1)}\\
{{\partial _u}(3uv)}&{{\partial _v}(3uv)}
\end{array}} \right)
=\left( {\begin{array}{*{20}{c}}
1&2\\
{3v}&{3u}
\end{array}} \right)
\end{align}\
$$
Dg(f(0)) = Dg(1,2) = \left( {\begin{array}{*{20}{c}}
1&2\\
{3 \cdot 2}&{3 \cdot 1}
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
1&2\\
6&3
\end{array}} \right)$$
$$Df(0) = \left( {\begin{array}{*{20}{c}}
1&2&3\\
0&1&1
\end{array}} \right)$$\
$$D(g \circ f) = Dg \circ Df$$
$$Dg(f(0))Df(0) = \left( {\begin{array}{*{20}{c}}
1&2\\
6&3
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
1&2&3\\
0&1&1
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
1&4&5\\
6&{15}&{21}
\end{array}} \right)$$