Dimension $1$ case: Suppose $f,g:\mathbb R\to \mathbb R$ are smooth functions that fixes $0$. If $g'(0)\ne0$, then we can write $f(x)=\phi(x)g(x)$ locally in a neighborhood of $0$. This is simple because we can let $\phi=f/g$ which is smooth at $0$ by Taylor theorem.
I want to generalize this to higher dimension: Suppose Suppose $f,g:\mathbb R^n\to \mathbb R^n$ are smooth maps that fixes $0$, and $\det(Dg)\ne 0$. Can we write
$$f_i(x)=\sum_{j=1}^n\phi_{ij}(x)g_j(x)$$
locally for some smooth functions $\phi_{ij}$ near $0$?