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