Let the charges be placed on the x-axis, centered at the origin. The charge $+q$ is at $x=l/2$ and $-q$ is at $x=-l/2$. The dipole moment is $p=ql$.

For point A on the axial line at $(r, 0)$: The distances from A to $+q$ and $-q$ are $r_+ = r - l/2$ and $r_- = r + l/2$, respectively. The net electric field at A is the superposition of the fields from each charge:

$$E_A = E_+ - E_- = kq\left( \frac{1}{(r-l/2)^2} - \frac{1}{(r+l/2)^2} \right)$$ $$E_A = kq \frac{(r+l/2)^2 - (r-l/2)^2}{(r^2 - l^2/4)^2} = kq \frac{2rl}{(r^2 - l^2/4)^2}$$

For $r \gg l$, we can neglect the $l^2/4$ term compared to $r^2$ in the denominator.

$$E_A \approx kq \frac{2rl}{(r^2)^2} = \frac{2kql}{r^3}$$

For point B on the perpendicular bisector (equatorial line) at $(0, r)$: The distance from either charge to B is $d = \sqrt{r^2 + (l/2)^2}$. The magnitude of the electric field from each charge at B is $E_+ = E_- = E = \frac{kq}{d^2} = \frac{kq}{r^2+l^2/4}$. By symmetry, the field components perpendicular to the dipole axis (y-components) cancel out. The components parallel to the axis (x-components) add up. Let $\theta$ be the angle between the x-axis and the line connecting a charge to point B. Then $\cos\theta = (l/2)/d$. The net field $E_B$ is the sum of the x-components:

$$E_B = E_+ \cos\theta + E_- \cos\theta = 2E\cos\theta$$ $$E_B = 2 \left( \frac{kq}{r^2+l^2/4} \right) \left( \frac{l/2}{\sqrt{r^2+l^2/4}} \right) = \frac{kql}{(r^2+l^2/4)^{3/2}}$$

For $r \gg l$, we can neglect the $l^2/4$ term compared to $r^2$.

$$E_B \approx \frac{kql}{(r^2)^{3/2}} = \frac{kql}{r^3}$$