Answer
$E = \frac{p}{2\pi~\epsilon_0~r^3}$
Work Step by Step
We can state Equation (24-30):
$V = \frac{1}{4\pi~\epsilon_0}~\frac{p~cos~\theta}{r^2}$
If the point is on the dipole axis, then $\theta = 0$
We can find an expression for $E$:
$E = -\frac{dV}{dr}$
$E = -\frac{-2}{4\pi~\epsilon_0}~\frac{p~cos~0^{\circ}}{r^3}$
$E = \frac{p}{2\pi~\epsilon_0~r^3}$
