Let $f(x) = \large\frac{ax+b}{px^2+qx+r}$

By qoutient rule,

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

$ = \Large\frac{(px^2+qx+r)(a)-(ax+b)(2px+q)}{(px^2+qx+r)^2}$

$ = \Large\frac{apx^2+aqx+ar-2apx^2-aqx-2bpx-bq}{(px^2+qx+r)^2}$

$ = \Large\frac{-apx^2-2bpx+ar-bq}{(bx^2+qx+r)}$