Question

(ax + b)ⁿ (cx + d)ⁿ 

Answer 1

\[\left( ax + b \right)^n \left( cx + d \right)^n \]
\[\text{ Let } u = \left( ax + b \right)^n , v = \left( cx + d \right)^n \]
\[\text{ Then }, u' = na \left( ax + b \right)^{n - 1} , v' = nc \left( cx + d \right)^{n - 1} \]
\[\text{ Using the product rule }: \]
\[\frac{d}{dx}\left( uv \right) = uv' + u'v\]
\[\frac{d}{dx}\left[ \left( ax + b \right)^n \left( cx + d \right)^n \right] = \left( ax + b \right)^n \times nc \left( cx + d \right)^{n - 1} + na \left( ax + b \right)^{n - 1} \times \left( cx + d \right)^n \]
\[ = n \left( ax + b \right)^{n - 1} \left( cx + d \right)^{n - 1} \left( acx + cb + acx + ad \right)\]
\[ = n \left( ax + b \right)^{n - 1} \left( cx + d \right)^{n - 1} \left( 2acx + cb + ad \right)\]
\[\]