Question

\[\frac{p x^2 + qx + r}{ax + b}\]

Answer 1

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