Assuming $U=\{x\in\mathbb{R}^2:x_1^2+x_2^2<1\}$ is the open unit circle in the plane and $f,g:U\rightarrow\mathbb{R}^2$ two functions with $f(0)=g(0)=0$.
$f$ is Fréchet-differentiable in $0$, and $\large\frac{\partial g}{\partial x_1}$, $\large\frac{\partial g}{\partial x_2}$ exist in $0$.
Does that imply that both $f\circ g$, $g\circ f$ are Fréchet-differentiable in $0$? And do both $f\circ g$, $g\circ f$ have $\large\frac{\partial f\circ g}{\partial x_1}$ and $\large\frac{\partial g\circ f}{\partial x_2}$in $0$?