Differential of a two variable's inverse functoin

Question

Let the function $f: \mathbb{R^2}\rightarrow\mathbb{R^2}$ be one-to-one and $C^1$, and g be its inverse ($f\circ g=id$). We know that the derivation of a linear map is itself. Namely, $D(f\circ g)(x,y)=D(id)(x,y)=id(x,y)$, which is a $2*1$ matrix. On the other hand, by chain rule: $D(f\circ g)(x,y)= Df(g(x,y))\cdot Dg(x,y)$, which will give us a $2*2$ matrix.
I am sure the second result (i.e. $2*2$ matrix) is right, but I cannot see where I'm making a mistake.

Answer 1

$D(\text{id})(x,y)=\text{id}(x,y)$ and the $\text{id}$ is interpreted as the matrix $(a_{i,j})_{1\leq i,j\leq 2}$, where $a_{1,1}=1$, $a_{1,2}=0$, $a_{2,1}=0$, and $a_{2,2}=1$.