KCET · Maths · Differentiation
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 \(-x \sin x^{2}\)
- C \(-2 x \sin 2 x^{2}\)
- D \(-x \cos 2 x^{2}\)
Answer & Solution
Correct Answer
(C) \(-2 x \sin 2 x^{2}\)
Step-by-step Solution
Detailed explanation
\(y=\left(\cos x^{2}\right)^{2}\)
On differentiating w.r.t. \(x\), we get
\(\begin{aligned}
\frac{d y}{d x} &=2 \cos x^{2} \frac{d}{d x}\left(\cos x^{2}\right) \\
&=2 \cos x^{2} \times\left(-\sin x^{2}\right) \frac{d}{d x}\left(x^{2}\right) \\
&=-4 x \cos x^{2} \sin x^{2} \\
&=-2 x \sin 2 x^{2}
\end{aligned}\)
On differentiating w.r.t. \(x\), we get
\(\begin{aligned}
\frac{d y}{d x} &=2 \cos x^{2} \frac{d}{d x}\left(\cos x^{2}\right) \\
&=2 \cos x^{2} \times\left(-\sin x^{2}\right) \frac{d}{d x}\left(x^{2}\right) \\
&=-4 x \cos x^{2} \sin x^{2} \\
&=-2 x \sin 2 x^{2}
\end{aligned}\)
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