Let $f(x,y)=(x-y, x+y) $ and $\varphi \in S (\mathbb R^2).$ Prove that $\varphi \circ f \in S(\mathbb R^2 ),$ where $S $ is the Schwartz space and $(x,y)\in \mathbb R^2.$
I tried to apply
$ \forall \alpha, \beta \in \mathbb N^d$
$|(x^{\alpha} \ D^{\beta}( \varphi \circ f)(X)$|= $|(x^{\alpha} \ D^{\beta} \varphi \circ f(X). D^{\beta}f(X)|$= $|(x^{\alpha} \ D^{\beta} \varphi (f(X)). Df(X)|$
If $\beta >1$ then $D^{\beta}f(X)=0$.
Any hint will be appreciated.