The given mathematical expression is:

$y = \sqrt[3]{\frac{8}{x}}$

To find the derivative, follow these steps:

Step 1: Rewrite the expression

Using the property of exponents: $y = \left( \frac{8}{x} \right)^{\frac{1}{3}}$

Step 2: Apply the chain rule

The derivative of $y = u^{\frac{1}{3}}$ is: $\frac{dy}{dx} = \frac{1}{3} u^{-\frac{2}{3}} \cdot \frac{du}{dx}$

Here, $u = \frac{8}{x}$.

Step 3: Differentiate $u = \frac{8}{x}$

The derivative of $\frac{8}{x}$ is: $\frac{du}{dx} = -\frac{8}{x^2}$

Step 4: Combine the results

Substitute $u = \frac{8}{x}$ and $\frac{du}{dx} = -\frac{8}{x^2}$ into the derivative: $\frac{dy}{dx} = \frac{1}{3} \left( \frac{8}{x} \right)^{-\frac{2}{3}} \cdot \left( -\frac{8}{x^2} \right)$

Step 5: Simplify

Simplify the expression step by step: $\frac{dy}{dx} = -\frac{8}{3} \cdot \left( \frac{8}{x} \right)^{-\frac{2}{3}} \cdot \frac{1}{x^2}$

Rewriting the final derivative: $\frac{dy}{dx} = -\frac{8}{3x^2} \cdot \left( \frac{x}{8} \right)^{\frac{2}{3}}$

This is the simplified derivative. Would you like further explanation or any assistance with specific steps?

Expanding Understanding:

1. Why is the chain rule used in this problem?
2. How does simplifying negative exponents work?
3. What are other interpretations of roots and fractional exponents?
4. How does this result behave as $x \to 0$ or $x \to \infty$?
5. Can this result be visualized graphically?

Tip: Practice rewriting roots and fractions as powers to make differentiation easier!