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We are tasked with finding the derivative of the function:

\[
f(x) = \sqrt[3]{(x^2 - 2)^2}
\]

and evaluating it at \( x = -1 \).

### Step-by-Step Solution:

1. **Rewrite the Function**:
   The function can be rewritten in a more manageable form:
   \[
   f(x) = \left( (x^2 - 2)^2 \right)^{1/3}
   \]
   This helps us prepare for the chain rule.

2. **Differentiate Using the Chain Rule**:
   The derivative of \( f(x) \) involves multiple applications of the chain rule. We'll first differentiate the outer function, which is a power of \( 1/3 \), and then differentiate the inner function.
   
   Let:
   \[
   g(x) = (x^2 - 2)^2
   \]
   Then \( f(x) = g(x)^{1/3} \).

   The derivative of \( g(x)^{1/3} \) with respect to \( x \) is:
   \[
   f'(x) = \frac{1}{3} g(x)^{-2/3} \cdot g'(x)
   \]
   
3. **Differentiate \( g(x) = (x^2 - 2)^2 \)**:
   We apply the chain rule to find \( g'(x) \).

   \[
   g'(x) = 2(x^2 - 2) \cdot \frac{d}{dx}(x^2 - 2)
   \]
   \[
   g'(x) = 2(x^2 - 2) \cdot 2x = 4x(x^2 - 2)
   \]

4. **Substitute \( g(x) \) and \( g'(x) \) into the Derivative**:
   Now, substituting \( g(x) = (x^2 - 2)^2 \) and \( g'(x) = 4x(x^2 - 2) \) into the expression for \( f'(x) \):
   \[
   f'(x) = \frac{1}{3} \cdot (x^2 - 2)^{-4/3} \cdot 4x(x^2 - 2)
   \]
   Simplify:
   \[
   f'(x) = \frac{4x(x^2 - 2)}{3(x^2 - 2)^{4/3}} = \frac{4x}{3(x^2 - 2)^{1/3}}
   \]

5. **Evaluate at \( x = -1 \)**:
   Substitute \( x = -1 \) into the derivative:
   \[
   f'(-1) = \frac{4(-1)}{3((-1)^2 - 2)^{1/3}} = \frac{-4}{3(1 - 2)^{1/3}} = \frac{-4}{3(-1)^{1/3}}
   \]
   The cube root of \( -1 \) is \( -1 \), so:
   \[
   f'(-1) = \frac{-4}{3(-1)} = \frac{4}{3}
   \]

Thus, the derivative of the function at \( x = -1 \) is:

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

Would you like further details or have any additional questions?

Here are 5 questions that could further expand on this:

1. What is the general form of the chain rule?
2. How does the power rule apply when dealing with fractional exponents?
3. Why do we need to apply the product rule when differentiating \( g'(x) = 4x(x^2 - 2) \)?
4. How can we simplify expressions involving cube roots when solving derivatives?
5. What are the implications of differentiating at a negative value like \( x = -1 \)?

**Tip**: When dealing with nested functions, always break down the differentiation step by step using the chain rule to avoid errors.

Find the derivative of the function f(x) = ( (x^2 - 2)^2 )^(1/3) at x = -1.

This page demonstrates how to find the derivative of the function f(x) = ( (x^2 - 2)^2 )^(1/3) at x = -1 using the chain rule and power rule. Detailed steps show the differentiation process and evaluation at the given point.

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