# How do you use the Product Rule to find the derivative of y = x^2 (x^2 + 1)^5?

Aug 15, 2015

$\frac{\mathrm{dy}}{\mathrm{dx}} = 2 x {\left({x}^{2} + 1\right)}^{4} \left(6 {x}^{2} + 1\right)$

#### Explanation:

The product rule states:
$f = g h \implies f ' = g ' h + g h '$

For this function we have:
$f \left(x\right) = y , g \left(x\right) = {x}^{2} , h \left(x\right) = {\left({x}^{2} + 1\right)}^{5}$
$g ' \left(x\right) = 2 x , h ' \left(x\right) = 10 x {\left({x}^{2} + 1\right)}^{4}$

Therefore by the charnel we have:
$\frac{\mathrm{dy}}{\mathrm{dx}} = g ' h + g h ' = 2 x {\left({x}^{2} + 1\right)}^{5} + 10 {x}^{3} {\left({x}^{2} + 1\right)}^{4} = 2 x {\left({x}^{2} + 1\right)}^{4} \left(6 {x}^{2} + 1\right)$