To find the derivative of $y = \sqrt[3]{x}$ using the definition of the derivative, we use the following limit definition:

$f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$

Here, $f(x) = \sqrt[3]{x}$. Thus, the derivative $f'(x)$ is given by:

[ f'(x) = \lim_{\Delta x \to 0} \frac{\sqrt[3]{x + \Delta x} - \sqrt[3]{x}}{\Delta x} ]

Let's solve this step by step.

Step 1: Write the expression for the difference quotient

[ f'(x) = \lim_{\Delta x \to 0} \frac{\sqrt[3]{x + \Delta x} - \sqrt[3]{x}}{\Delta x} ]

Step 2: Multiply and divide by the conjugate

To simplify the expression, multiply the numerator and the denominator by the conjugate of the numerator, which in this case involves the expression that eliminates the cube root in the numerator. We multiply by:

[ \frac{\sqrt[3]{(x + \Delta x)^2} + \sqrt[3]{(x + \Delta x)x} + \sqrt[3]{x^2}}{\sqrt[3]{(x + \Delta x)^2} + \sqrt[3]{(x + \Delta x)x} + \sqrt[3]{x^2}} ]

This step can be quite complex, but it simplifies the expression by eliminating the cube roots.

Step 3: Simplify the expression

After multiplying by the conjugate, the numerator simplifies to:

[ (\sqrt[3]{x + \Delta x} - \sqrt[3]{x}) \times (\sqrt[3]{(x + \Delta x)^2} + \sqrt[3]{(x + \Delta x)x} + \sqrt[3]{x^2}) = \Delta x ]

So, we have:

[ f'(x) = \lim_{\Delta x \to 0} \frac{\Delta x}{\Delta x \left( \sqrt[3]{(x + \Delta x)^2} + \sqrt[3]{(x + \Delta x)x} + \sqrt[3]{x^2} \right)} ]

This simplifies to:

[ f'(x) = \lim_{\Delta x \to 0} \frac{1}{\sqrt[3]{(x + \Delta x)^2} + \sqrt[3]{(x + \Delta x)x} + \sqrt[3]{x^2}} ]

Step 4: Take the limit as $\Delta x \to 0$

As $\Delta x \to 0$, the expression inside the cube root simplifies, and we get:

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

Thus, the derivative of $y = \sqrt[3]{x}$ is:

$\boxed{f'(x) = \frac{1}{3x^{2/3}}}$

Would you like more details or have any other questions?

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Here are five related questions you might find interesting:

1. How would you find the derivative of $y = \sqrt[3]{x^2}$ using the definition of the derivative?
2. What is the derivative of $y = \sqrt[3]{x}$ using standard differentiation rules?
3. How can you use the power rule to differentiate $y = x^{1/3}$?
4. How does the process differ when finding the derivative of a square root function instead of a cube root?
5. What are the common errors when applying the definition of the derivative to functions with roots?

Tip: When dealing with cube roots, multiplying by the conjugate helps to simplify the expression by eliminating the root.