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