# How do you use the Product Rule to find the derivative of y = (x^2 + 6) (4x^6 + 5) ?

Aug 16, 2015

$\textcolor{red}{\frac{\mathrm{dy}}{\mathrm{dx}} = 24 {x}^{5} \left({x}^{2} + 6\right) + 2 x \left(4 {x}^{6} + 5\right)}$

#### Explanation:

The Product Rule, states that if $y = u v$, then

$\frac{\mathrm{dy}}{\mathrm{dx}} = u \frac{\mathrm{dv}}{\mathrm{dx}} + v \frac{\mathrm{du}}{\mathrm{dx}}$

Let $u = {x}^{2} + 6$ and $v = 4 {x}^{6} + 5$

Then

$\frac{\mathrm{du}}{\mathrm{dx}} = 2 x$

$\frac{\mathrm{dv}}{\mathrm{dx}} = 24 {x}^{5}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = u \frac{\mathrm{dv}}{\mathrm{dx}} + v \frac{\mathrm{du}}{\mathrm{dx}}$

dy/dx=(x^2+6)× 24x^5 +(4x^6+5)×2x

$\frac{\mathrm{dy}}{\mathrm{dx}} = 24 {x}^{5} \left({x}^{2} + 6\right) + 2 x \left(4 {x}^{6} + 5\right)$