To find the derivative of $\sqrt[3]{x^4}$, we can rewrite the expression using exponent notation and then apply the chain rule.

Step-by-step process:

1. Rewrite $\sqrt[3]{x^4}$ as $(x^4)^{1/3}$, which simplifies to $x^{4/3}$.
2. Use the power rule to differentiate $x^{n}$, where $n = \frac{4}{3}$. The power rule states that the derivative of $x^n$ is ( n x^{n-1}