Let $f(x) = \large\frac{4x+5 \sin x}{3x+7 \cos x}$

By quotient rule,

$ f'(x) = \large\frac{(3x+7 \cos x) \large\frac{d}{dx}(4x+5\sin x)-(4x+5\sin x) \large\frac{d}{dx}(3x+7\cos x)}{(3x+7 \cos x)^2}$

$ =\large\frac{ (3x+7\cos x) \bigg[4 \large\frac{d}{dx}(x)+5 \large\frac{d}{dx}(\sin x)\bigg]-(4x+5 \sin x) \bigg[3 \large\frac{d}{dx}(x)+7 \large\frac{d}{dx}(\cos x)\bigg]}{(3x+7\cos x)^2}$

$ = \large\frac{(3x+7\cos x)(4+5 \cos x)-(4x+5\sin x)(3-7\sin x)}{(3x+7 \cos x)^2}$

$ = \large\frac{12x+ 15x\cos x+28 \cos x+35\cos^2x-12x+28x\sin x-15\sin x+35\sin^2x}{(3x+7\cos x^2)}$

$ = \large\frac{ 15x\cos x+28 \cos x+28x\sin x-15\sin x+35(\cos^2x\sin^2x)}{(3x+7\cos x^2)}$

$ = \large\frac{ 35+15x\cos x+28 \cos x+28x\sin x-15\sin x}{(3x+7\cos x^2)}$