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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):
$\frac{x^{2} \cos \left(\frac{\pi}{4}\right)}{\sin x}$
Solution:
Let $f(x)=\frac{x^{2} \cos \left(\frac{\pi}{4}\right)}{\sin x}$
By quotient rule,
$f^{\prime}(x)=\cos \frac{\pi}{4} \cdot\left[\frac{\sin x \frac{d}{d x}\left(x^{2}\right)-x^{2} \frac{d}{d x}(\sin x)}{\sin ^{2} x}\right]$
$=\cos \frac{\pi}{4} \cdot\left[\frac{\sin x \cdot 2 x-x^{2} \cos x}{\sin ^{2} x}\right]$
$=\frac{x \cos \frac{\pi}{4}[2 \sin x-x \cos x]}{\sin ^{2} x}$
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