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Question
\[\frac{ax + b}{p x^2 + qx + r}\]
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Solution
\[\text{ Let } u = ax + b; v = p x^2 + qx + r\]
\[\text{ Then }, u' = a; v' = 2px + q\]
\[\text{ Using the quotient rule }:\]
\[\frac{d}{dx}\left( \frac{u}{v} \right) = \frac{vu' - uv'}{v^2}\]
\[\frac{d}{dx}\left( \frac{ax + b}{p x^2 + qx + r} \right) = \frac{\left( p x^2 + qx + r \right)a - \left( ax + b \right)\left( 2px + q \right)}{\left( p x^2 + qx + r \right)^2}\]
\[ = \frac{ap x^2 + aq x + ar - 2ap x^2 - 2bp x - aq x - bq}{\left( p x^2 + qx + r \right)^2}\]
\[ = \frac{- ap x^2 - 2bp x + ar - bq}{\left( p x^2 + qx + r \right)^2}\]
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