Given the electric potential of a dipole

$$\phi(r, \theta)=\frac{p \cos \theta}{4 \pi \epsilon_{0} r^{2}},$$

where $p$ is the magnitude of the dipole moment, calculate the corresponding electric field and show that it can be written as

$$\mathbf{E}(\mathbf{r})=\frac{1}{4 \pi \epsilon_{0}} \frac{1}{r^{3}}\left[3\left(\mathbf{p} \cdot \hat{\mathbf{e}}_{r}\right) \hat{\mathbf{e}}_{r}-\mathbf{p}\right]$$

where $\hat{\mathbf{e}}_{r}$ is the unit vector in the radial direction.