How do you differentiate  f(x)=(2x+1)(4-x^2)(1+x^2)  using the product rule?

Oct 12, 2016

Answer:

$10 {x}^{4} + 4 {x}^{3} - 18 {x}^{2} - 6 x - 8$

You could further simply it if you are required to do so.

Explanation:

Multiply $\left(1 + {x}^{2}\right)$ by $\left(4 - {x}^{2}\right)$: $\left(4 + 3 {x}^{2} - {x}^{4}\right)$

$\left(\left(2 x + 1\right) ' \left(4 + 3 {x}^{2} - {x}^{4}\right)\right) + \left(\left(2 x + 1\right) \left(4 + 3 {x}^{2} - {x}^{4}\right) '\right)$

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

$\left(8 + 6 {x}^{2} - 2 {x}^{4}\right) + \left(6 x + 12 {x}^{2} - 4 {x}^{3} - 8 {x}^{4}\right)$

$8 + 6 {x}^{2} - 2 {x}^{4} + 6 x + 12 {x}^{2} - 4 {x}^{3} - 8 {x}^{4}$

$8 + 6 x + 18 {x}^{2} - 4 {x}^{3} - 10 {x}^{4}$

Optional, divide by $- 1$,

$10 {x}^{4} + 4 {x}^{3} - 18 {x}^{2} - 6 x - 8$

Good Luck