KCET · Maths · Differentiation
If \(y=\sin ^{n} x \cos n x\), then \(\frac{d y}{d x}\) is
- A \(n \sin ^{n-1} x \sin (n+1) x\)
- B \(n \sin ^{n-1} x \cos (n-1) x\)
- C \(n \sin ^{n-1} x \cos n x\)
- D \(n \sin ^{n-1} x \cos (n+1) x\)
Answer & Solution
Correct Answer
(D) \(n \sin ^{n-1} x \cos (n+1) x\)
Step-by-step Solution
Detailed explanation
Given, \(\mathrm{y}=\sin ^{\mathrm{n}} \mathrm{x} \cos \mathrm{nx}\)
\[
\begin{aligned}
\frac{d y}{d x} &=n \sin ^{n-1} x \cos x \cos n x-n \sin ^{n} x \sin n x \\
&=n \sin ^{n-1} x[\cos x \cos n x-\sin x \sin n x] \\
&=n \sin ^{n-1} x \cos (n+1) x
\end{aligned}
\]
\[
\begin{aligned}
\frac{d y}{d x} &=n \sin ^{n-1} x \cos x \cos n x-n \sin ^{n} x \sin n x \\
&=n \sin ^{n-1} x[\cos x \cos n x-\sin x \sin n x] \\
&=n \sin ^{n-1} x \cos (n+1) x
\end{aligned}
\]
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