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

Mar 14, 2018

$\left(4 x + 4\right) \left(5 {x}^{3} + 2 x + 2\right) + \left(15 {x}^{2} + 2\right) \left(2 {x}^{2} + 4 x - 3\right)$

$50 {x}^{4} + 80 {x}^{3} - 33 {x}^{2} + 24 x + 2$

#### Explanation:

First the product rule is, $g \left(x\right) = f p r i m e \left(x\right) h \left(x\right) + h p r i m e \left(x\right) f \left(x\right)$

Where $f \left(x\right) = 2 {x}^{2} + 4 x - 3$
And $h \left(x\right) = 5 {x}^{3} + 2 x + 2$

Now take the derivative of both, this gives you...
$f p r i m e \left(x\right) = \left(4 x + 4\right)$
$h p r i m e \left(x\right) = \left(15 {x}^{2} + 2\right)$

So now plug into the product rule formula
$\left(4 x + 4\right) \left(5 {x}^{3} + 2 x + 2\right) + \left(15 {x}^{2} + 2\right) \left(2 {x}^{2} + 4 x - 3\right)$

After multiplying and adding like terms you get
$50 {x}^{4} + 80 {x}^{3} - 33 {x}^{2} + 24 x + 2$