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 nonzero constants and m and n are integers): (ax + b) (cx + d)2

Answer 1

Let f(x)=(ax+b)(cx+d)2
By Leibnitz product rule,
f'(x)=(ax+b) $\frac{d}{dx}$(cx+d)2+(cx+d)2$\frac{d}{dx}$(ax+b)
= (ax+b)$\frac{d}{dx}$(c2x2+2cdx+d2)+(cx+d)2$\frac{d}{dx}$(ax+b)
= (ax+b) [$\frac{d}{dx}$(c2x2) +$\frac{d}{dx}$(2cdx)+$\frac{d}{dx}$ d2)]+(cx+d)2[$\frac{d}{dx}$ ax+$\frac{d}{dx}$ b]
=(ax+b) (2c2x+2cd)+(cx+d2)a
=2c(ax+b) (cx+d)+a(cx+d)2