How do you differentiate f(x)= (2x+1)(x^3+x^2)  using the product rule?

Jul 29, 2016

$f ' \left(x\right) = x \left(8 {x}^{2} + 9 x + 2\right)$

Explanation:

The product rule states that

$\frac{d}{\mathrm{dx}} \left(f \left(x\right) g \left(x\right)\right) = f ' \left(x\right) g \left(x\right) + f \left(x\right) g ' \left(x\right)$

$f \left(x\right) = 2 x + 1 \mathmr{and} g \left(x\right) = {x}^{3} + {x}^{2}$

$\implies f ' \left(x\right) = 2 \mathmr{and} g ' \left(x\right) = 3 {x}^{2} + 2 x$

$f ' \left(x\right) g \left(x\right) + f \left(x\right) g ' \left(x\right) = 2 {x}^{3} + 2 {x}^{2} + \left(2 x + 1\right) \left(3 {x}^{2} + 2 x\right)$

$= 2 {x}^{3} + 2 {x}^{2} + 6 {x}^{3} + 4 {x}^{2} + 3 {x}^{2} + 2 x = 8 {x}^{3} + 9 {x}^{2} + 2 x$

$= x \left(8 {x}^{2} + 9 x + 2\right)$