Question

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

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

Answer 1

The correct option is D

$\sqrt{\mathrm{\pi }}$

Find the $\frac{dy}{dx}$:

Given that, $y=\log \left[\sin \left(x^2\right)\right],0<x<\frac{\pi }{2}$

Differentiate $y$ with respect to $x$.

$\begin{array}{rcl}\frac{dy}{dx}& =& \left[\frac{1}{\sin \left(x^2\right)}\right]\cos \left(x^2\right)2x\\ & =& 2x\cot \left(x^2\right)\end{array}$
Put $x=\frac{\sqrt{\pi }}{2}$
$\begin{array}{rcl}\frac{\mathrm{dy}}{\mathrm{dx}}& =& 2\left(\frac{\sqrt{\mathrm{\pi }}}{2}\right)\cot \frac{\mathrm{\pi }}{4}\\ & =& \sqrt{\mathrm{\pi }}\end{array}$ \\
Hence option D is correct.