To find the derivative of the function $f(x) = \sqrt[3]{\frac{8}{x}}$, let's rewrite it in a form that's easier to differentiate.

1. Rewrite the expression:

   $f(x) = \left( \frac{8}{x} \right)^{\frac{1}{3}}$

2. Rewrite using power laws:

   Since $\frac{8}{x} = 8x^{-1}$, we can write:

   $f(x) = \left( 8x^{-1} \right)^{\frac{1}{3}} = 8^{\frac{1}{3}} x^{-\frac{1}{3}}$

   Since $8^{\frac{1}{3}} = 2$, we get:

   $f(x) = 2x^{-\frac{1}{3}}$

3. Differentiate $f(x) = 2x^{-\frac{1}{3}}$:

   Using the power rule $\frac{d}{dx} x^n = nx^{n-1}$, we get:

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

4. Simplify:

   $f'(x) = -\frac{2}{3} x^{-\frac{4}{3}}$

5. Rewrite in terms of radicals:

   Since $x^{-\frac{4}{3}} = \frac{1}{x^{\frac{4}{3}}}$, the derivative is:

   $f'(x) = -\frac{2}{3x^{\frac{4}{3}}}$

Final Answer:

$f'(x) = -\frac{2}{3x^{\frac{4}{3}}}$

Would you like further clarification on any of the steps, or details on how the power rule was applied?

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1. What is the chain rule and how does it apply to derivatives of roots?
2. How do we rewrite radicals as exponents for easier differentiation?
3. What are alternative methods for differentiating functions with fractional exponents?
4. How does the power rule for derivatives apply to negative exponents?
5. What are real-life applications of derivatives involving roots?

Tip: For fractional powers, always try to rewrite the expression as a power of $x$ to make differentiation straightforward.