Aloha :)
Hier empfehle hier die Verwendung des Nabla-Kalküls:
$$\text{div}\,\left(r^n\cdot\vec r\right)=\nabla\left(r^n\cdot\vec r\right)=\left(\nabla r^n\right)\cdot\vec r+r^n\nabla\vec r$$$$=\left(\frac{\partial(r^n)}{\partial r}\cdot\frac{\partial r}{\partial\vec r}\right)\cdot\vec r+r^n\left(\begin{array}{c}\partial_x\\\partial_y\\\partial_z\end{array}\right)\cdot\left(\begin{array}{c}x\\y\\z\end{array}\right)$$$$=\left(nr^{n-1}\cdot\left(\begin{array}{c}\partial_x\\\partial_y\\\partial_z\end{array}\right)\sqrt{x^2+y^2+z^2}\right)\vec r+r^n(1+1+1)$$$$=\left(nr^{n-1}\cdot\frac{1}{2\sqrt{x^2+y^2+z^2}}\left(\begin{array}{c}2x\\2y\\2z\end{array}\right)\right)\vec r+3r^n$$$$=\left(nr^{n-1}\cdot\frac{\vec r}{r}\right)\vec r+3r^n=nr^n+3r^n=(n+3)r^n$$
