1. **Problem statement:** Find the derivative $\frac{dy}{dx}$ of the function $y = f(x) = \sqrt[3]{x^2}$. 2. **Rewrite the function:** Recall that $\sqrt[3]{x^2} = x^{\frac{2}{3}}$. 3. **Formula for derivative:** Use the power rule for derivatives: if $y = x^n$, then $\frac{dy}{dx} = n x^{n-1}$. 4. **Apply the power rule:** Here, $n = \frac{2}{3}$, so $$\frac{dy}{dx} = \frac{2}{3} x^{\frac{2}{3} - 1} = \frac{2}{3} x^{-\frac{1}{3}}.$$ 5. **Rewrite the derivative:** Since $x^{-\frac{1}{3}} = \frac{1}{x^{\frac{1}{3}}} = \frac{1}{\sqrt[3]{x}}$, we have $$\frac{dy}{dx} = \frac{2}{3 \sqrt[3]{x}}.$$ 6. **Final answer:** $$\boxed{\frac{dy}{dx} = \frac{2}{3 \sqrt[3]{x}}}.$$