Q98 / 180 · KCET 2021 · Mathematics

If $y=\left(\cos x^{2}\right)^{2}$, then $\frac{d y}{d x}$ is equal to

  • A
    $-4 x \sin 2 x^{2}$
  • B
    $-x \sin x^{2}$
  • Correct answer
    C
    $-2 x \sin 2 x^{2}$
  • D
    $-x \cos 2 x^{2}$

Solution

$y=\left(\cos x^{2}\right)^{2}$ On differentiating w.r.t. $x$, we get $\begin{aligned} \frac{d y}{d x} &=2 \cos x^{2} \frac{d}{d x}\left(\cos x^{2}\right) \\ &=2 \cos x^{2} \times\left(-\sin x^{2}\right) \frac{d}{d x}\left(x^{2}\right) \\ &=-4 x \cos x^{2} \sin x^{2} \\ &=-2 x \sin 2 x^{2} \end{aligned}$

Hence, the correct answer is option: 'C'.

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