How do you differentiate y=(-7x^4+10x^(2/5)+8)(x^2+10) using the product rule?

Oct 5, 2017

$\frac{\mathrm{dy}}{\mathrm{dx}} = \left(- 7 {x}^{4} + 10 {x}^{\frac{2}{5}} + 8\right) 2 x + \left({x}^{2} + 10\right) \left(- 28 {x}^{3} + 4 {x}^{- \frac{3}{5}}\right)$

Explanation:

Solve by using the product rule.
$\frac{\mathrm{dy}}{\mathrm{dx}} = f \left(x\right) g ' \left(x\right) + g \left(x\right) f ' \left(x\right)$

$f \left(x\right) = - 7 {x}^{4} + 10 {x}^{\frac{2}{5}} + 8$
$g \left(x\right) = {x}^{2} + 10$
$f ' \left(x\right) = 28 {x}^{3} + 4 {x}^{- \frac{3}{5}}$
$g ' \left(x\right) = 2 x$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \left(- 7 {x}^{4} + 10 {x}^{\frac{2}{5}} + 8\right) 2 x + \left({x}^{2} + 10\right) \left(- 28 {x}^{3} + 4 {x}^{- \frac{3}{5}}\right)$