$$f(x, y, z)=\left(\frac{x^2}{x^2+y^2+z^2},\frac{y^2}{x^2+y^2+z^2},\frac{z^2}{x^2+y^2+z^2}\right)$$
Show that $f$ is locally invertible at every point except the origin and find an explicit formula for the inverse.
I can show that the determinant of the derivative matrix is not $0$ when the point is not in the origin, but I am stuck on how to find an explicit formula.
I know that in this case, the derivative of the inverse is the inverse of the derivative, but how does that work in multivariable?