Prove that Kelvin Transform is Harmonic in Spherical Coordinates

Question

I have been struggling for the past couple of days to demonstrate that the Kelvin transform: $$ w(x) = |x|^{2-n}u\left(\frac{x}{|x|^2}\right) $$ is harmonic given that u(x) is harmonic. I am interested in showing this for only n=3, and I have tried to utilize spherical coordinates (a brute force computational approach is desired). My difficulty is in considering the second order partial derivatives of $u(x/|x|^2)$. I understand that this is a composite function, but I am unsure of how to consider the partial derivatives here. I would appreciate if somebody could guide me through this in spherical coordinates, or at least point me towards the necessary formula/theorem or literature for attempting this.