To find the derivative of the function \( f(x) = (x^3 + 1)^3(x - 1)^2 \), we'll apply the **product rule** and the **chain rule**.

### Step 1: Identify the components
Let:
- \( u(x) = (x^3 + 1)^3 \)
- \( v(x) = (x - 1)^2 \)

### Step 2: Apply the product rule
The product rule states:
\[
f'(x) = u'(x)v(x) + u(x)v'(x)
\]

### Step 3: Find the derivatives of \( u(x) \) and \( v(x) \)
- **Derivative of \( u(x) \):**
  
  Use the chain rule for \( u(x) = (x^3 + 1)^3 \):
  \[
  u'(x) = 3(x^3 + 1)^2 \cdot \frac{d}{dx}(x^3 + 1)
  \]
  \[
  u'(x) = 3(x^3 + 1)^2 \cdot 3x^2 = 9x^2(x^3 + 1)^2
  \]

- **Derivative of \( v(x) \):**
  
  For \( v(x) = (x - 1)^2 \):
  \[
  v'(x) = 2(x - 1) \cdot \frac{d}{dx}(x - 1) = 2(x - 1) \cdot 1 = 2(x - 1)
  \]

### Step 4: Substitute into the product rule
Now, substitute \( u'(x) \), \( u(x) \), \( v(x) \), and \( v'(x) \) into the product rule:
\[
f'(x) = 9x^2(x^3 + 1)^2 \cdot (x - 1)^2 + (x^3 + 1)^3 \cdot 2(x - 1)
\]

### Step 5: Simplify the expression (if needed)
The derivative can be simplified, but the expression provided is the derivative:
\[
f'(x) = 9x^2(x^3 + 1)^2(x - 1)^2 + 2(x - 1)(x^3 + 1)^3
\]

This is the derivative of the function \( f(x) \).

Would you like more details on any step, or do you have any questions?

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**Related Questions:**
1. How do you apply the chain rule in differentiation?
2. Can you differentiate a product of more than two functions?
3. How does the derivative change if the exponents in the function are different?
4. What are some common mistakes when applying the product rule?
5. How do you find the second derivative of a product function?

**Tip:** Always simplify each part of the function before applying the product or chain rule, as it makes the differentiation process more manageable.

Find the derivative of the function f(x) = (x³ + 1)³(x - 1)² using the product rule and chain rule. Detailed step-by-step solution includes identifying the components, applying differentiation rules, and simplifying the expression. Suitable for students in Grades 11-12 or early college calculus.

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