# How do you use the quotient rule to find the derivative of y=(x-sqrt(x))/(x^(1/3)) ?

Sep 8, 2014

$y ' = \frac{2}{3} \cdot \frac{1}{x} ^ \left(\frac{1}{3}\right) - \frac{1}{6} \cdot \frac{1}{x} ^ \left(\frac{5}{6}\right)$

Solution:

y=(x−sqrtx)/x^(1/3)

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

$y = {x}^{\frac{2}{3}} - {x}^{\frac{1}{6}}$

differentiating with respect to $x$,

$y ' = \frac{2}{3} {x}^{\frac{2}{3} - 1} - \frac{1}{6} {x}^{\frac{1}{6} - 1}$

$y ' = \frac{2}{3} {x}^{- \frac{1}{3}} - \frac{1}{6} {x}^{- \frac{5}{6}}$

$y ' = \frac{2}{3} \cdot \frac{1}{x} ^ \left(\frac{1}{3}\right) - \frac{1}{6} \cdot \frac{1}{x} ^ \left(\frac{5}{6}\right)$