To differentiate $y = 3 \sqrt[3]{x}$, we will first rewrite the expression in exponent form:

$y = 3 x^{\frac{1}{3}}$

Now, we can apply the power rule for differentiation:

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

Here, $n = \frac{1}{3}$. So, applying the power rule:

$\frac{dy}{dx} = 3 \cdot \frac{1}{3} x^{\frac{1}{3} - 1} = x^{-\frac{2}{3}}$

Thus, the derivative is:

$\frac{dy}{dx} = x^{-\frac{2}{3}}$

Or equivalently:

$\frac{dy}{dx} = \frac{1}{x^{\frac{2}{3}}}$

Do you need further details or have any questions?

Here are 5 related questions:

1. How can we differentiate other fractional powers of $x$?
2. What happens if we have a negative exponent in the function?
3. How do we handle the differentiation of a product of functions?
4. How can we use the chain rule to differentiate composite functions?
5. What is the application of the derivative in real-life problems?

Tip: Always rewrite radicals as exponents when differentiating to make applying the power rule easier!