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Find the derivative of the following functions

Question:

Find the derivative of the following functions (it is to be understood that abcdp, q, r and s are fixed non-zero constants and m and n are integers): $\frac{\sin x+\cos x}{\sin x-\cos x}$

 

Solution:

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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