Explanation
Concept:
cos2x=2cos2x−1
sin2x=2sinxcosx
Calculation:
Here, y=cos2x2
Let, x2=t
Differentiating with respect to x, we get
⇒ 2xdx=dt
⇒ dt/dx=2x ...(1)
y=cos2t
\begin{aligned}
& =\frac{\cos2\mathrm{t}+1}{2}=\frac{\cos2\mathrm{t}}{2}+\frac{1}{2} \\
& \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{2}\frac{\mathrm{d}}{\mathrm{dt}}\left(\cos2\mathrm{t}\right)\frac{\mathrm{dt}}{\mathrm{dx}}+0 \\
& =\frac{1}{2}(-2\sin2\mathrm{t})\frac{\mathrm{dt}}{\mathrm{dx}}\cdots(\mathrm{from}(1)) \\
& =-\sin2\mathbf{x}^{2}\times2\mathbf{x} \\
& =-4\mathrm{x}\cos\mathrm{x}^{2}\sin\mathrm{x}^{2} \\
& \mathrm{Hence,~option~(2)~is~correct.}
\end{aligned}
