Let's find the derivative of the given function step by step:

Given:
$y = \frac{x^3}{\sqrt[3]{x}}$

First, express the cube root of $x$ as a power of $x$. Recall that $\sqrt[3]{x} = x^{1/3}$. So the function becomes:

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

Now, simplify the expression by subtracting exponents (since $\frac{a^m}{a^n} = a^{m-n}$):

$y = x^{3 - \frac{1}{3}} = x^{\frac{9}{3} - \frac{1}{3}} = x^{\frac{8}{3}}$

Now, differentiate with respect to $x$ using the power rule $\frac{d}{dx}[x^n] = n x^{n-1}$:

$\frac{dy}{dx} = \frac{8}{3} x^{\frac{8}{3} - 1} = \frac{8}{3} x^{\frac{5}{3}}$

Thus, the derivative is:

$\frac{dy}{dx} = \frac{8}{3} x^{\frac{5}{3}}$

So, the derivative of $y$ is $\frac{8}{3} x^{\frac{5}{3}}$.

This result differs from the expression you provided, which is $\frac{8}{3} x \sqrt[3]{x^2}$. However, both forms are equivalent. The expression $x^{\frac{5}{3}}$ is the same as $x \sqrt[3]{x^2}$, because $x^{\frac{5}{3}} = x \cdot x^{\frac{2}{3}} = x \sqrt[3]{x^2}$.

Do you want more details or have any further questions on this? Here's a tip for you:

Tip: Always remember to simplify expressions before differentiating, especially when dealing with roots and powers.