If $y =$ log (sin($x^{2})),\ 0

Question

If $y =$ log (sin($x^{2})),\ 0 < x < \frac{\pi}{2},\ then\frac{dy}{dx}$ at $x = \frac{\sqrt{\pi}}{2}$ is

Dr. Abubacker · Accepted Answer

(d)
Given, $y = \log{(\sin{(x^{2}))}}$
On differentiating w.r.t. $x,\$we get
$\frac{dy}{dx} = \frac{1}{\sin x^{2}}\ .\cos{x^{2}.2x = 2x\cot x^{2}}$
At $x = \frac{\sqrt{\pi}}{2},\frac{dy}{dx} = \frac{2\sqrt{\pi}}{2}\cot\left( \frac{\sqrt{\pi}}{2} \right)^{2}$
$= \sqrt{\pi}\cot{\left( \frac{\pi}{4} \right) = \sqrt{\pi}}$

Q. If $y =$ log (sin($x^{2})),\ 0 < x < \frac{\pi}{2},\ then\frac{dy}{dx}$ at $x = \frac{\sqrt{\pi}}{2}$ is

Explanation

Reviewed by Dr. Abubacker

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