Given $f_1, f_2$ be $C^1$-vector fields on $\mathbb{R}^n$, we define the lie bracket
$$ [f_1,f_2](x)=\frac{df_2}{dx}(x)f_1(x)-\frac{df_1}{dx}(x)f_2(x)$$
They are said to commute if the lie bracket is identically $0$.
We need to show that if $f_1, f_2\in C^1(\mathbb{R}^n)$ are commuting and $f_1(0)=f_2(0)=0$, then the Jacobian matrices $Df_1(0)$ and $Df_2(0)$ commute. I have no idea where to begin. A hint will be appreciated.