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Question
\[\frac{x^5 - \cos x}{\sin x}\]
Solution
\[\text{ Let } u = x^5 - \cos x; v = \sin x\]
\[\text{ Then }, u' = 5 x^4 + \sin x; v' = \cos x\]
\[\text{ Using the quotient rule }:\]
\[\frac{d}{dx}\left( \frac{u}{v} \right) = \frac{vu' - uv'}{v^2}\]
\[\frac{d}{dx}\left( \frac{x^5 - \cos x}{\sin x} \right) = \frac{\sin x\left( 5 x^4 + \sin x \right) - \left( x^5 - \cos x \right)\cos x}{\sin^2 x}\]
\[ = \frac{- x^5 \cos x + 5 x^4 \sin x + \left( \sin^2 x + \cos^2 x \right)}{\sin^2 x}\]
\[ = \frac{- x^5 \cos x + 5 x^4 \sin x + 1}{\sin^2 x}\]
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