Sobolev-Gagliardo-Nirenberg: Why is $|f|^q$ continously differentiable?

Question

I wanna understand a proof of the Sobolev-Gagliardo-Nirenberg inequality. Therefore, I need to know why $|f|^q \in C_c^1(\mathbb{R}^n)$ for $f \in C_c^1(\mathbb{R}^n)$ and $q>1$. Can eventually someone tell me why this holds?

Answer 1

The function $\varphi(x) = |x|^q$ is continuously differentiable on $\mathbf R$ for all $q > 1$ and you can calculate its derivative explicitly: $$\varphi'(x) = \left\{ \begin{array}{ll} qx |x|^{q-2} & x \not= 0 \\ 0 & x=0 \end{array} \right.$$

Now use the chain rule.