find the differentiation

Put costheta = (1- tan²theta/2 )/(1 + tan²theta/2)

So on solving we get sin^-1 [ {(a+b) - (b-a)tan²theta/2}/{(a+b) + (b-a)tan²theta/2}]

which is pie/2 - 2tan^-1[root(b-a)tantheta/2]

so we can easily find dy/dx = -root(b-a).sec²(theta/2) / ( 1 + (b-a)tan²theta/2 )