Question

(ax + b)ⁿ (cx + d)ⁿ 

Answer 1

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