Let $\phi: \mathbb{R}^n \rightarrow \mathbb{R}$ be a $C^\infty$ function with the following properties.
$\phi(x) = 1$ if $|x| \leq 1$
$\phi(x) = 0$ if $|x| \geq 2$
$0 \leq \phi \leq 1$
$\phi$ is radial.
Let $\phi_k(x) = \phi(\frac{x}{k})$. How do I show that for each multiindex $\alpha$, there exists a constant $C_\alpha$ such that $|D^\alpha \phi_k| \leq \frac{C_\alpha}{k^{\textrm{deg }\alpha}}$ uniformly in $k$?