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): $(a x+b)(c x+d)^{2}$
Solution:
Let $f(x)=(a x+b)(c x+d)^{2}$
By Leibnitz product rule,
$f^{\prime}(x)=(a x+b) \frac{d}{d x}(c x+d)^{2}+(c x+d)^{2} \frac{d}{d x}(a x+b)$
$=(a x+b) \frac{d}{d x}\left(c^{2} x^{2}+2 c d x+d^{2}\right)+(c x+d)^{2} \frac{d}{d x}(a x+b)$
$=(a x+b)\left[\frac{d}{d x}\left(c^{2} x^{2}\right)+\frac{d}{d x}(2 c d x)+\frac{d}{d x} d^{2}\right]+(c x+d)^{2}\left[\frac{d}{d x} a x+\frac{d}{d x} b\right]$
$=(a x+b)\left(2 c^{2} x+2 c d\right)+\left(c x+d^{2}\right) a$
$=2 c(a x+b)(c x+d)+a(c x+d)^{2}$