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Let's solve this problem where \( y = \sqrt[3]{x} \) and find \( y' \).

### Step 1: Express the function in exponential form
The cube root of \( x \) can be written as:
\[
y = \sqrt[3]{x} = x^{\frac{1}{3}}
\]

### Step 2: Differentiate with respect to \( x \)
To find \( y' \), differentiate both sides with respect to \( x \):
\[
\frac{dy}{dx} = \frac{d}{dx} \left( x^{\frac{1}{3}} \right)
\]

### Step 3: Apply the power rule
The power rule for differentiation is \( \frac{d}{dx} \left( x^n \right) = n \cdot x^{n-1} \). Applying this to our function:
\[
y' = \frac{1}{3} \cdot x^{\frac{1}{3} - 1} = \frac{1}{3} \cdot x^{-\frac{2}{3}}
\]

### Step 4: Simplify the expression
\[
y' = \frac{1}{3} \cdot \frac{1}{x^{\frac{2}{3}}} = \frac{1}{3} x^{-\frac{2}{3}}
\]

This corresponds to one of the choices provided.

The correct answer is:
\[
\boxed{\frac{1}{3} x^{-\frac{2}{3}}}
\]

Would you like more details or have any questions? Here are some related questions:

1. How does the power rule apply to fractional exponents?
2. What is the derivative of \( y = \sqrt[4]{x} \)?
3. How can we find the derivative of a more complex root function like \( y = \sqrt[5]{x^2} \)?
4. How does negative exponent notation simplify understanding derivatives?
5. What happens if you differentiate the function \( y = \frac{1}{x} \)?
6. How do you differentiate products or quotients of functions?
7. Can you provide a physical interpretation of the derivative for this function?
8. What is the process for finding higher-order derivatives of \( y = \sqrt[3]{x} \)?

**Tip:** When dealing with roots, always express them as fractional exponents to simplify the differentiation process.

Learn how to differentiate y = cube root of x to find y'. Step-by-step solution using the power rule for differentiation. Suitable for advanced high school students and those studying calculus.

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