How do you differentiate g(x) = (2x^2 + 4x - 3) ( 2x + 2) using the product rule?

Feb 25, 2016

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

Explanation:

differentiate using the$\textcolor{b l u e}{\text{ Product rule }}$

If $g \left(x\right) = f \left(x\right) \cdot h \left(x\right)$, then $g ' \left(x\right) = f \left(x\right) \cdot h ' \left(x\right) + h \left(x\right) \cdot f ' \left(x\right)$

let $f \left(x\right) = 2 {x}^{2} + 4 x - 3$

$\Rightarrow f ' \left(x\right) = 4 x + 4 = 4 \left(x + 1\right)$

and let $h \left(x\right) = 2 x + 2 \Rightarrow h ' \left(x\right) = 2$

substitute values back into $g ' \left(x\right)$

hence $g ' \left(x\right) = \left[\left(2 {x}^{2} + 4 x - 3\right) \cdot 2\right] + \left[\left(2 x + 2\right) \cdot 4 \left(x + 1\right)\right]$

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

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

If you like, this can be simplified into a single polynomial expression:

$g ' \left(x\right) = 12 {x}^{2} + 24 x + 2$