# How do you differentiate f(x) = (2x+1)(4-x^2)(1+x^2) ?

Oct 5, 2015

$\frac{\mathrm{df} \left(x\right)}{\mathrm{dx}} = - 10 {x}^{4} - 4 {x}^{3} + 18 {x}^{2} + 6 x + 8$

#### Explanation:

$f \left(x\right) = \left(8 x + 4 - 2 {x}^{3} - {x}^{2}\right) \left(1 + {x}^{2}\right)$
$f \left(x\right) = 8 x + 8 {x}^{3} + 4 + 4 {x}^{2} - 2 {x}^{3} - 2 {x}^{5} - {x}^{2} - {x}^{4}$
$f \left(x\right) = - 2 {x}^{5} - {x}^{4} + 6 {x}^{3} + 3 {x}^{2} + 8 x + 4$

$\frac{\mathrm{df} \left(x\right)}{\mathrm{dx}} = - 10 {x}^{4} - 4 {x}^{3} + 18 {x}^{2} + 6 x + 8$

Oct 5, 2015

The product rule for 3 factors is: $\frac{d}{\mathrm{dx}} \left(u v w\right) = u ' v w + u v ' w + u v w '$

#### Explanation:

So

$f ' \left(x\right) = \left(2\right) \left(4 - {x}^{2}\right) \left(1 + {x}^{2}\right) + \left(2 x + 1\right) \left(2 x\right) \left(1 + {x}^{2}\right) + \left(2 x + 1\right) \left(4 - {x}^{2}\right) \left(2 x\right)$

Simplify algebraically as you see fit.