Let $f(x) = \large\frac{ax+b}{cx+d}$

By quotient rule,

$f'(x) =\large\frac{(cx+d) \Large\frac{d}{dx}(ax+b) -(ax+b)\Large\frac{d}{dx}(cx+d)}{(cx+d)^2}$

$ = \large\frac{(cx+d)(a)-(ax+b)(c)}{(cx+d)^2}$

$ = \large\frac{acx+ad-acx-bc}{(cx+d)^2}$

$ = \large\frac{ad-bc}{(cx+d)^2}$