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