Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): $\frac{\sin x+\cos x}{\sin x-\cos x}$
Let $f(x)=\frac{\sin x+\cos x}{\sin x-\cos x}$
By quotient rule,
$f^{\prime}(x)=\frac{(\sin x-\cos x) \frac{d}{d x}(\sin x+\cos x)-(\sin x+\cos x) \frac{d}{d x}(\sin x-\cos x)}{(\sin x-\cos x)^{2}}$
$=\frac{(\sin x-\cos x)(\cos x-\sin x)-(\sin x+\cos x)(\cos x+\sin x)}{(\sin x-\cos x)^{2}}$
$=\frac{-(\sin x-\cos x)^{2}-(\sin x+\cos x)^{2}}{(\sin x-\cos x)^{2}}$
$=\frac{-\left[\sin ^{2} x+\cos ^{2} x-2 \sin x \cos x+\sin ^{2} x+\cos ^{2} x+2 \sin x \cos x\right]}{(\sin x-\cos x)^{2}}$
$=\frac{-[1+1]}{(\sin x-\cos x)^{2}}$
$=\frac{-2}{(\sin x-\cos x)^{2}}$
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