Question

Let $y=\log \sin \left(x^2\right),0<x\le \frac{\pi }{2}$. The value of $\frac{dy}{dx}$ at $x=\frac{\sqrt{\pi }}{2}$ is:

• A. $0$
• B. $1$
• C. $\frac{\pi }{-4}$
• D. $\sqrt{\pi }$

Answer 1

The correct option is D $\sqrt{\pi }$

$y=\log \sin \left(x^2\right)$

Chain rule leads us to $\frac{dy}{dx}=\frac{1}{\sin \left(x^2\right)}\times \cos \left(x^2\right)\times 2x$

Substituting $x$ as $\frac{\sqrt{\pi }}{2}$, we get $\frac{dy}{dx}=\cot \frac{\pi }{4}\times \sqrt{\pi }=\sqrt{\pi }$