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Question
If `y = (sin x + cos x)/(sin x - cos x)`, then `(dy)/(dx)` at x = 0 is ______.
Options
–2
0
`1/2`
Does not exist
Solution
If `y = (sin x + cos x)/(sin x - cos x)`, then `(dy)/(dx)` at x = 0 is –2.
Explanation:
Given `y = (sinx + cosx)/(sinx - cosx)`
`(dy)/(dx) = ((sin x - cos x)(cos x - sin x) - (sin x + cos x)(cos x + sin x))/(sin x - cos x)^2`
= `(-(sin x - cos x)^2 - (sin x + cos x)^2)/(sin x - cos x)^2`
= `(-[sin^2x + cos^2x - 2 sinx cosx + sin^2x + cos^2x + 2 sinx cos x])/(sin x - cos x)62`
= `(-2)/(sin x - cos x)^2`
∴ `((dy)/(dx))_("at" x = 0) = (-2)/(sin 0 - cos 0)^2`
= `(-2)/(-1)^2`
= –2
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