Chapter 3 - Differentiation - 3.4 Rates of Change - Exercises - Page 129: 3

$f'(1) = \frac{1}{3}$ $f'(8) = \frac{1}{12}$ $f'(27) = \frac{1}{27}$

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: $$f'(1) = \frac{1}{3}(1)^{-2/3} = \frac{1}{3}$$ $$f'(8) = \frac{1}{3}(8)^{-2/3} = \frac{1}{3}*\frac{1}{4} = \frac{1}{12}$$ $$f'(27) = \frac{1}{3}(27)^{-2/3}= \frac{1}{3}*\frac{1}{9} = \frac{1}{27}$$

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