How do you differentiate y = (sqrt x + (1/2))(x^3 + x^(1/3))?

Oct 14, 2015

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{7 {x}^{\frac{5}{2}} + 3 {x}^{2}}{2} + \frac{5 {x}^{4} + {x}^{- \frac{2}{3}}}{6}$

Explanation:

The easiest way is to foil and then derivate

$y = \left({x}^{\frac{1}{2}} + \frac{1}{2}\right) \left({x}^{3} + {x}^{1} / 3\right)$

$y = {x}^{3 + \frac{1}{2}} + {x}^{3} / 2 + {x}^{\frac{1}{3} + \frac{1}{2}} + {x}^{\frac{1}{3}} / 2$

$y = {x}^{\frac{7}{2}} + {x}^{3} / 2 + {x}^{\frac{5}{6}} + {x}^{\frac{1}{3}} / 2$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{7 {x}^{\frac{5}{2}}}{2} + \frac{3 {x}^{2}}{2} + \frac{5 {x}^{4}}{6} + \frac{{x}^{- \frac{2}{3}}}{6}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{7 {x}^{\frac{5}{2}} + 3 {x}^{2}}{2} + \frac{5 {x}^{4} + {x}^{- \frac{2}{3}}}{6}$