Question

\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\] 

Answer 1

\[\text{ Let } u = a x^2 + bx + c; v = p x^2 + qx + r\]
\[\text{ Then }, u' = 2ax + b; v' = 2px + q\]
\[\text{ Using the quotient rule }:\]
\[\frac{d}{dx}\left( \frac{u}{v} \right) = \frac{vu' - uv'}{v^2}\]
\[\frac{d}{dx}\left( \frac{a x^2 + bx + c}{p x^2 + qx + r} \right) = \frac{\left( p x^2 + qx + r \right)\left( 2ax + b \right) - \left( a x^2 + bx + c \right)\left( 2px + q \right)}{\left( p x^2 + qx + r \right)^2}\]
\[ = \frac{2ap x^3 + 2aq x^2 + 2arx + bp x^2 + bqx + br - 2ap x^3 - 2bp x^2 - 2pcx - aq x^2 - bqx - cq}{\left( p x^2 + qx + r \right)^2}\]
\[ = \frac{\left( aq - bp \right) x^2 + 2\left( ar - xp \right)x + br - cq}{\left( p x^2 + qx + r \right)^2}\]