Answer
$f'(1) = \frac{1}{3}$
$f'(8) = \frac{1}{12}$
$f'(27) = \frac{1}{27}$
Work Step by Step
Let $f(x) = \sqrt[3] x = x^{1/3}$.
We first need to find $f'(x)$ using the power rule:
$f'(x) = \frac{1}{3}x^{-2/3}$
Now, to find the rate of change at the three points, we just substitute x into the equation above to get:
$$f'(1) = \frac{1}{3}(1)^{-2/3} = \frac{1}{3}$$
$$f'(8) = \frac{1}{3}(8)^{-2/3} = \frac{1}{3}*\frac{1}{4} = \frac{1}{12}$$
$$f'(27) = \frac{1}{3}(27)^{-2/3}= \frac{1}{3}*\frac{1}{9} = \frac{1}{27}$$
