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If y = log (sin (x^(2))), 0 lt x lt (pi...

If ` y = log (sin (x^(2))), 0 lt x lt (pi)/(2), "then " (dy)/(dx) "at x " = (sqrt(pi))/(2) ` is

A

0

B

1

C

`(pi)/(4)`

D

`sqrt(pi)`

Text Solution

Verified by Experts

The correct Answer is:
D
Given , ` y = log (sin (x^(2))`
On differentiating both sides of Eq .(i) w.r.t.x, we get
` (dy)/(dx) = (1)/(sin x^(2)) . cos x^(2) - 2x = 2x cot x^(2)`
` At x = (sqrt(pi))/(2) ,(dy)/(dx) = (2sqrt(pi))/(2)cot (sqrt(pi)/(2))^(2)= sqrt(pi) cot ((pi)/(4))= sqrt(pi)`
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