Q83 / 180 · KCET 2009 · Mathematics

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
    $n \sin ^{n-1} x \cos n x$
  • Correct answer
    D
    $n \sin ^{n-1} x \cos (n+1) x$

Solution

Given, $\mathrm{y}=\sin ^{\mathrm{n}} \mathrm{x} \cos \mathrm{nx}$ $$ \begin{aligned} \frac{d y}{d x} &=n \sin ^{n-1} x \cos x \cos n x-n \sin ^{n} x \sin n x \\ &=n \sin ^{n-1} x[\cos x \cos n x-\sin x \sin n x] \\ &=n \sin ^{n-1} x \cos (n+1) x \end{aligned} $$

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

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