Consider the an electric dipole of charges +q and −q separated by distance 2a with center at O. 

Goal: To find electric field at point P on the axial line of the dipole, OP=r. 

Let E1​ and E2​ be electric field on P due to charges +q and −q respectively. 

E1​=(r−a)2kq​ along AP 

E2​=(r+a)2kq​ along BP

The resultant electric field at P, E=E1​−E2​ (as both E1​ and E2​ are in opposite direction) 

E=(r−a)2kq​−(r+a)2kq​=kq(r2−a2)24ra​

Define,  p=2aq 

E=k(r2−a2)22pr​

If r>>a, then E=kr42pr​=kr32p​. 

In vector form, E=kr32p​​.