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Question
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
Solution
\[\frac{d}{dx}\left( \frac{\left( x^3 + 1 \right)\left( x - 2 \right)}{x^2} \right)\]
\[ = \frac{d}{dx}\left( \frac{x^4 - 2 x^3 + x - 2}{x^2} \right)\]
\[ = \frac{d}{dx}\left( \frac{x^4}{x^2} \right) - 2\frac{d}{dx}\left( \frac{x^3}{x^2} \right) + \frac{d}{dx}\left( \frac{x}{x^2} \right) - \frac{d}{dx}\left( \frac{2}{x^2} \right)\]
\[ = \frac{d}{dx}\left( x^2 \right) - 2\frac{d}{dx}\left( x \right) + \frac{d}{dx}\left( x^{- 1} \right) - 2\frac{d}{dx}\left( x^{- 2} \right)\]
\[ = 2x - 2 - \frac{1}{x^2} - 2\left( - 2 \right) x^{- 3} \]
\[ = 2x - 2 - \frac{1}{x^2} + \frac{4}{x^3}\]
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