Answer
$f'(x) = \frac{2x}{3(x^{2}- 1)^{\frac{2}{3}}}$
$f'(3) = \frac{1}{2}$
Work Step by Step
1. Find the derivative
$f(x) = \sqrt[3] {x^{2} -1 }$
$f(x) = (x^{2}- 1)^{\frac{1}{3}}$
$f'(x) = \frac{1}{3}(x^{2}- 1)^{-\frac{2}{3}}(2x)$
$f'(x) = \frac{2x}{3(x^{2}- 1)^{\frac{2}{3}}}$
2. Evaluate the derivative at point $(3,2 )$
$f'(3) = \frac{2x}{3(x^{2}- 1)^{\frac{2}{3}}}$
$= \frac{2(3)}{3((3)^{2}- 1)^{\frac{2}{3}}}$
$= \frac{6}{3(9- 1)^{\frac{2}{3}}}$
$= \frac{6}{3(8)^{\frac{2}{3}}}$
$= \frac{6}{3(4)}$
$= \frac{6}{12}$
$= \frac{1}{2}$
