In the lecture on Gagliardo-Niremberg inquality, there was mentioned the fact that for functions: $$f_{\lambda}(x) = (\lambda + \vert x \vert^q)^\frac{p-n}{p}, \; \text{ where } \frac{1}{q}+\frac{1}{p}=1$$ equality holds.
On checking this, I got stuck on calculating the gradient $\nabla f$. Knowing that $f$ is radially symmetric, is there a property I can use?