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If \(\cfrac{\pi}2\) ≤ x ≤ \(\cfrac{3\pi}2\) and y = sin-1(sin x), find \(\cfrac{dy}{d\mathrm x} \).

1 Answer

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Best answer

For x ∈ \((\cfrac{\pi}2,\cfrac{3\pi}2)\)

y=sin-1 (sin x) = sin-1 ( sin (π –(π -x))

(to get y in principal range of sin-1 x)

i.e.,

y = π - x

\(\therefore \cfrac{dy}{d\mathrm x} \) = -1

From the last problem we see that

So, y is not differentiable at x = \(\cfrac{\pi}2.\)

Extending this, we can say that y is not differentiable at 

x = (2n + 1)\(\cfrac{\pi}2\)

So, for x ∈ \(\Big[\cfrac{\pi}2,\cfrac{3\pi}2\Big]\)

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