Mathematics - Functions Question with Solution | TestHub

On differentiating (i) w.r.t.x, we get
$f^{\prime }(x)=\frac{-2\sin x^2\cos x^2}{\sqrt{1+{\cos }^2{x}^2}}(x)$

Put, $x=\frac{\sqrt{\pi }}{2}$ in (ii), we get
$\begin{array}{rl}& {\mathrm{f}}^{\prime }\left(\frac{\sqrt{\pi }}{2}\right)=-\frac{\sqrt{\pi }}{2}\cdot \frac{\sin 2\left(\frac{\pi }{4}\right)}{\sqrt{1+\frac{1}{2}}}\\ & =-\frac{\sqrt{\pi }}{2}\cdot \frac{\sin \frac{\pi }{2}}{\sqrt{\frac{3}{2}}}=-\sqrt{\frac{\pi }{6}}\end{array}$