The derivative of the given function $f(x) = \sqrt[3]{(x^2 - 2)^2}$ at $x = -1$ is:

$f'(-1) = \frac{4}{3}$

Would you like further details on this process? Here are five related questions for further exploration:

1. How do you differentiate functions involving radicals and powers?
2. What is the Chain Rule, and how does it apply here?
3. How can the derivative be verified using numerical methods?
4. How would the result change if we evaluated the derivative at a different point?
5. How can implicit differentiation be used in similar cases?

Tip: When differentiating functions involving roots, consider rewriting them as powers for easier manipulation.