Let $f(x) =\large\frac{\sin x+\cos x}{\sin x - \cos x}$

By quotient rule,

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

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

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

$ = \large\frac{-[ \sin^x+\cos^2x-2\sin x \cos x + \sin^2x+\cos^2x+2\sin x \cos x]}{(\sin x - \cos x )^2}$

$ = \large\frac{ -[1+1]}{(\sin x - \cos x)^2}$

$ = \large\frac{ -2}{(\sin x - \cos x)^2}$