$\bigtriangledown r^n=\bigg(\hat{i} \frac {\partial}{\partial x}+\hat{j} \frac {\partial}{\partial y} +\hat{k} \frac {\partial}{\partial z}\bigg)r^n\ =\hat{i} \bigg(nr^{n-1}\frac {\partial r}{\partial x}\bigg)+ \hat{j} \bigg(nr^{n-1}\frac {\partial r}{\partial y}\bigg) +\hat{k} \bigg(nr^{n-1}\frac {\partial r}{\partial z}\bigg)……………….(1)\ |\vec{r}|=\sqrt {x^2+y^2+z^2}\ or\ r^2=x^2+y^2+z^2………………………….(2)\ Differentiating(2)\ partially\ with\ respect\ to\ x,\ y\ and\ z,\ we\ get:\ 2r\frac {\partial}{\partial x}=2x\ or\ \frac {\partial}{\partial x}=\frac {x}{r},2r\frac {\partial}{\partial y}=2y\ or\ \frac {\partial}{\partial y}=\frac {y}{r},\ and\ 2r\frac {\partial}{\partial z}=2z\ or\ \frac {\partial}{\partial z}=\frac {z}{r}\ Substituting \ the\ values\ in\ equation(1),\ we\ get\ \bigtriangledown r^n =\hat{i} \bigg(nr^{n-1}\frac {x}{r}\bigg)+ \hat{j} \bigg(nr^{n-1}\frac {y}{r}\bigg) +\hat{k} \bigg(nr^{n-1}\frac {z}{r}\bigg)\ \bigtriangledown r^n =nr^{n-2}(x\hat{i}+ y\hat{j}+ z\hat{k})= nr^{n-2}\vec{r}. $