Prove that $f:\mathbf{R}^2\to \mathbf{R}$ given by $f(x,y)=e^{-(x^2+y^2)^{-1}}$ is $C^\infty$, and give the Taylor polynomial of order $n$ in $\vec{0}$.
I need to show that all partial derivatives in $\vec{0}$ need to be zero, i.e. $D_1^{k_1}D_2^{k_2}f(0,0)=0$. From there, I don't know how to proceed. Any help would be helpful. Thanks!