$\dfrac {\vec{r}} {\| \vec{r} \|} = \left\{ \dfrac {x} {\sqrt{x^2+y^2+z^2}},\dfrac {y} {\sqrt{x^2+y^2+z^2}},\dfrac {z} {\sqrt{x^2+y^2+z^2}} \right \}$

Take the divergence and then take the gradient and plow through the algebra. It's not that bad. You end up with the right hand side when you're done.