Question

If $y=Sin\,x \cdot Sin\, 2x \cdot Sin\, 3 x \ldots Sin\, n \,x$, then y' is

• A. $\displaystyle\sum_{k=1}^{n} k\, tan\, k\,x$
• B. $y . \displaystyle\sum_{k=1}^{n} k\,cot\, kx$
• C. $y. \displaystyle\sum_{k=1}^{n} k\, tan\, kn$
• D. $\displaystyle\sum_{k=1}^{n} cot\, kx$

Answer 1

Answer: (B)

Given, $y=\sin x \cdot \sin 2 x \cdot \sin 3 x \cdot \ldots \sin n x$
Taking $\log$ on both sides,
$\log y=\log \sin x+\log \sin 2 x+\ldots+\log \sin n x$
Differentiating w.r.t. $x$
$\frac{1}{y} \cdot \frac{d y}{d x}=1 \cdot \cot x+2 \cot 2 x+\ldots +n \cot n x$
$\Rightarrow \frac{d y}{d x}= y \cdot \displaystyle \sum_{k=1}^{n} k \cot k x$
$\Rightarrow y'=y \cdot \displaystyle \sum_{k=1}^{n} k \cot k x$