Let $u: \mathbb{R}^{n} \longrightarrow \mathbb{R}$ be a smooth function, and for $x \in \mathbb{R}^{n} \backslash\{0\}$, define $$ K(u)(x):=\frac{u\left(x /|x|^{2}\right)}{|x|^{n-2}} $$ Show that $$ \Delta K(u)(x):=\frac{\Delta u\left(x /|x|^{2}\right)}{|x|^{n+2}} $$ In particular, if $\Delta u=0$, then show that $\Delta(K(u))=0$.
I have tried explicit calculation with the usual formula for the laplacian, that is the sum of the partial derivatives squared, but the expression I keep getting is unwieldy and nowhere close to the answer required. Any help would be appreciated.