Question

Find the derivative of the following functions (it is to be understood that $a,b,c,d,p,q,r$ and $s$ are fixed non-zero constants and $m$ and $n$ are integers) : $(ax+b)(cx+d{)}^2$

Answer 1

Let $f\left(x\right)=(ax+b)(cx+d{)}^2$
Thus by Leibnitz product rule
$f^{\prime }\left(x\right)=(ax+b)\frac{d}{dx}{(cx+d)}^2+{(cx+d)}^2\frac{d}{dx}(ax+b)$
$=(ax+b)\frac{d}{dx}(c^2{x}^2+2cdx+d^2)+{(cx+d)}^2\frac{d}{dx}(ax+b)$
$=(ax+b)[\frac{d}{dx}\left(c^2{x}^2\right)+\frac{d}{dx}\left(2cdx\right)+\frac{d}{dx}{d}^2]+{(cx+d)}^2[\frac{d}{dx}ax+\frac{d}{dx}b]$
$=(ax+b)(2c^{2}x+2cd)+(cx+d^2)a$
$=2a(ax+b)(cx+d)+a(cx+d{)}^2$