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Question
Find the derivative of the following function (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):
`(sin x + cos x)/(sin x - cos x)`
Solution
Let f(x) = `(sinx + cosx)/(sinx - cosx)`
∴ `f'(x) = ([d/dx(sin x + cos x)] (sin x - cos x) - (sin x +cos x) d/dx (sin x - cos x))/(sin x - cos x)^2`
= `((cos x - sin x)(sin x - cos x) - (sin x + cos x)(cos x + sin x))/(sin x - cos x)^2`
= `(-(cos x - sin x)^2 - (sin x + cos x)^2)/(sin x - cos x)^2`
= `(-(cos^2 x + sin^2 x - 2 cosx sinx) - (cos^2 x + sin^2 x + 2 sin x cos x))/(sin x - cosx)^2`
= `(1 - 2sin xcos x + 1 + 2 cosx sin x)/(sin x - cosx)^2`
= `(-2)/(sin x - cosx)^2`
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