Let $f _1$ be apparent frequency when source is moving towards observer $f _2$ be apparent frequency when source is moving away from observer Then change in frequency

$\Delta f = f _1- f _2$

$= f \left(\frac{ v }{ v - v _{ s }}\right)- f \left(\frac{ v }{ v + v _{ s }}\right)$

$= f \left[\left(1-\frac{ v _{ s }}{ v }\right)^{-1}-\left(1+\frac{ v _{ s }}{ v }\right)^{-1}\right]$

Using Binomial Theorem

$\approx f\left[\left(1+\frac{v_s}{v}\right)-\left(1-\frac{v_s}{v}\right)\right]=\frac{2 f v_s}{v}$