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

Apr 3, 2017

$y ' = - 4 {x}^{3} \left(9 {x}^{5} - 10 {x}^{4} + 15 x - 20\right)$

#### Explanation:

Denote $f \left(x\right) = {x}^{4} + 3$ and $g \left(x\right) = - 4 {x}^{5} + 5 {x}^{4} + 5$ so that $y$ is of the form $f . g$.

The derivative of $y$ with respect to $x$ according the product rule is then $f ' . g + f . g '$:

$y ' = f ' . g + f . g '$
$y ' = \left(4 {x}^{3}\right) \left(- 4 {x}^{5} + 5 {x}^{4} + 5\right) + \left({x}^{4} + 3\right) \left(- 20 {x}^{4} + 20 {x}^{3}\right)$
$y ' = - 16 {x}^{8} + 20 {x}^{7} + 20 {x}^{3} - 20 {x}^{8} + 20 {x}^{7} - 60 {x}^{4} + 60 {x}^{3}$
$y ' = - 36 {x}^{8} + 40 {x}^{7} - 60 {x}^{4} + 80 {x}^{3}$
$y ' = - 4 {x}^{3} \left(9 {x}^{5} - 10 {x}^{4} + 15 x - 20\right)$