To find the derivative of the function $f(x) = \sqrt[3]{x}$, we can rewrite the function as:

$f(x) = x^{1/3}$

Now, we apply the power rule of differentiation, which states that:

$\frac{d}{dx} \left( x^n \right) = n \cdot x^{n-1}$

Here, $n = \frac{1}{3}$, so:

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

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

$f'(x) = \frac{1}{3} \cdot x^{-\frac{2}{3}} = \frac{1}{3\sqrt[3]{x^2}}$

Would you like more details or have any questions about this process?

Here are 5 related questions for further practice:

1. What is the derivative of $f(x) = x^5$?
2. How do you find the derivative of $f(x) = \sqrt{x}$?
3. How can you apply the chain rule to differentiate $f(x) = \sqrt[3]{3x}$?
4. What is the second derivative of $f(x) = x^{1/3}$?
5. How do you differentiate a function with a negative exponent?

Tip: When differentiating roots, remember that the root can be expressed as a fractional exponent to simplify differentiation.