# How do you differentiate g(x) = (2x^2 + 4x - 3) ( 5x^3 + 2x + 2) using the product rule?

$g ' \left(x\right) = \frac{d}{\mathrm{dx}} g \left(x\right) = 50 {x}^{4} + 80 {x}^{3} - 33 {x}^{2} + 24 x + 2$

#### Explanation:

For derivative of product , we have the formula
$\frac{d}{\mathrm{dx}} \left(u v\right) = u \frac{\mathrm{dv}}{\mathrm{dx}} + v \frac{\mathrm{du}}{\mathrm{dx}}$

From the given $g \left(x\right) = \left(2 {x}^{2} + 4 x - 3\right) \left(5 {x}^{3} + 2 x + 2\right)$

We let $u = 2 {x}^{2} + 4 x - 3$ and $v = 5 {x}^{3} + 2 x + 2$

$\frac{d}{\mathrm{dx}} \left(g \left(x\right)\right) = \left(2 {x}^{2} + 4 x - 3\right) \frac{d}{\mathrm{dx}} \left(5 {x}^{3} + 2 x + 2\right) + \left(5 {x}^{3} + 2 x + 2\right) \frac{d}{\mathrm{dx}} \left(2 {x}^{2} + 4 x - 3\right)$

$\frac{d}{\mathrm{dx}} \left(g \left(x\right)\right) = \left(2 {x}^{2} + 4 x - 3\right) \left(15 {x}^{2} + 2\right) + \left(5 {x}^{3} + 2 x + 2\right) \left(4 x + 4\right)$

Expand to simplify

$\frac{d}{\mathrm{dx}} \left(g \left(x\right)\right) = \left(2 {x}^{2} + 4 x - 3\right) \left(15 {x}^{2} + 2\right) + \left(5 {x}^{3} + 2 x + 2\right) \left(4 x + 4\right)$

$\frac{d}{\mathrm{dx}} \left(g \left(x\right)\right) = 30 {x}^{4} + 4 {x}^{2} + 60 {x}^{3} + 8 x - 45 {x}^{2} - 6 + 20 {x}^{4} + 20 {x}^{3} + 8 {x}^{2} + 8 x + 8 x + 8$

Combine like terms

$\frac{d}{\mathrm{dx}} \left(g \left(x\right)\right) = 50 {x}^{4} + 80 {x}^{3} - 33 {x}^{2} + 24 x + 2$

God bless...I hope the explanation is useful.