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Question:
Find the derivative of the following functions (it is to be understood that $a, b, c, d, p, q, r$ and $s$ are fixed non-zero constants and $m$ and $n$ are integers): $\frac{x}{\sin ^{n} x}$
Solution:
Let $f(x)=\frac{x}{\sin ^{n} x}$
By quotient rule,
$f^{\prime}(x)=\frac{\sin ^{n} x \frac{d}{d x} x-x \frac{d}{d x} \sin ^{n} x}{\sin ^{2 n} x}$
It can be easily shown that $\frac{d}{d x} \sin ^{n} x=n \sin ^{n-1} x \cos x$
Therefore,
$f^{\prime}(x)=\frac{\sin ^{n} x \frac{d}{d x} x-x \frac{d}{d x} \sin ^{n} x}{\sin ^{2 n} x}$
$=\frac{\sin ^{n} x \cdot 1-x\left(n \sin ^{n-1} x \cos x\right)}{\sin ^{2 n} x}$
$=\frac{\sin ^{n-1} x(\sin x-n x \cos x)}{\sin ^{2 n} x}$
$=\frac{\sin x-n x \cos x}{\sin ^{n+1} x}$
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