Let $f(x) =\large\frac{a+b \sin x}{c+d \cos x}$

By quotient rule,

$f'(x) = \large\frac{(c+d \cos x)\large\frac{d}{dx}(a+b \sin x)-(a+b \sin x) \large\frac{d}{dx}(c+d \cos x)}{(c+d \cos x)^2}$

$ = \large\frac{(c+d \cos x)(b \cos x)-(a+b \sin x)(-d \sin x)}{(c+d \cos x)^2}$

$ = \large\frac{cb \cos x+bd \cos^2x+ad \sin x+bd \sin^2x}{(c+d \cos x)^2}$

$ = \large\frac{bc \cos x+ad \sin x+bd(\cos^2x+ \sin^2x)}{(c+d\cos x)^2}$

$ = \large\frac{bc \cos x+ad \sin x+bd}{(c+d \cos x)^2}$