# How do you use the product rule to differentiate g(x)=(x^2+1)(x^2-2x)?

Jan 14, 2017

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

#### Explanation:

$\text{Given " g(x)=f(x).h(x)" then}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{g ' \left(x\right) = f \left(x\right) h ' \left(x\right) + h \left(x\right) f ' \left(x\right)} \textcolor{w h i t e}{\frac{2}{2}} |}}} \leftarrow \text{ product rule}$

$\text{here } f \left(x\right) = {x}^{2} + 1 \Rightarrow f ' \left(x\right) = 2 x$

$\text{and } h \left(x\right) = {x}^{2} - 2 x \Rightarrow h ' \left(x\right) = 2 x - 2$

$\Rightarrow g ' \left(x\right) = \left({x}^{2} + 1\right) \left(2 x - 2\right) + \left({x}^{2} - 2 x\right) .2 x$

$= 2 {x}^{3} - 2 {x}^{2} + 2 x - 2 + 2 {x}^{3} - 4 {x}^{2}$

$= 4 {x}^{3} - 6 {x}^{2} + 2 x - 2$