Let the distance of object O from the lens is a image I1 is a distance v1, image I at distance v
From first surface AB image of O get on I1
\(\frac{\mu -1}{R_1} = \frac{\mu}{v_1} - \frac 1u\) ......(i)
From second surface CD image of I1 get
\(\frac{\frac 1\mu -1}{R_2} = \frac{\frac 1\mu -1}v - \frac 1{v_1}\) ......(ii)
\(\mu = \frac 1u, u = v_1\) for CD surface light ray passing from denser to rarer medium.
Equation (2) multiply by \(\mu\)
\(\frac 1v - \frac \mu{v_1} = - \frac{1 - \mu}{R_2}\) .....(iii)
adding equation (i) + (iii),
\(\frac 1v -\frac 1u = \frac{\mu -1}{R_1} + \frac{1-\mu}{R_2}\)
if \(u =\infty\)
\(v = f\) ......(iv)
\(\frac 1f = (\mu - 1)(\frac 1{R_1} - \frac 1{R_2})\) ......(v)
By comparing equation (iv) and (v)
\(\frac 1f = \frac 1v - \frac 1u\)