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Question
\[\text{ If } y = \left( \frac{2 - 3 \cos x}{\sin x} \right), \text{ find } \frac{dy}{dx} at x = \frac{\pi}{4}\]
Solution
\[\frac{dy}{dx} = \frac{d}{dx}\left( \frac{2 - 3 \cos x}{\sin x} \right)\]
\[ = \frac{d}{dx}\left( \frac{2}{\sin x} \right) - \frac{d}{dx}\left( \frac{3 \cos x}{\sin x} \right)\]
\[ = 2\frac{d}{dx}\left( \cos ec x \right) - 3\frac{d}{dx}\left( \cot x \right)\]
\[ = - 2 \cos ec x \cot x + 3 \cos e c^2 x\]
\[\frac{dy}{dx} at x=\frac{\pi}{4}= - 2 \cos ec \frac{\pi}{4} \cot \frac{\pi}{4} + 3 \cos e c^2 \frac{\pi}{4}\]
\[ = - 2\left( \sqrt{2} \right)\left( 1 \right) + 3 \left( \sqrt{2} \right)^2 \]
\[ = - 2\sqrt{2} + 6\]
\[ = 6 - 2\sqrt{2}\]
\[\]
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