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