To differentiate the function \( f(x) = (x^2 + 2x + 1)^3 \), we'll use the **chain rule**. The chain rule states that if you have a composite function \( g(h(x)) \), then the derivative is given by:

\[
\frac{d}{dx} [g(h(x))] = g'(h(x)) \cdot h'(x)
\]

### Step 1: Identify the outer and inner functions
- The outer function is \( g(u) = u^3 \), where \( u = x^2 + 2x + 1 \).
- The inner function is \( h(x) = x^2 + 2x + 1 \).

### Step 2: Differentiate the outer function
- The derivative of the outer function \( g(u) = u^3 \) with respect to \( u \) is \( g'(u) = 3u^2 \).

### Step 3: Differentiate the inner function
- The derivative of the inner function \( h(x) = x^2 + 2x + 1 \) with respect to \( x \) is \( h'(x) = 2x + 2 \).

### Step 4: Apply the chain rule
Using the chain rule:

\[
\frac{d}{dx} \left[ (x^2 + 2x + 1)^3 \right] = 3(x^2 + 2x + 1)^2 \cdot (2x + 2)
\]

### Final Answer:
\[
\frac{d}{dx} \left[ (x^2 + 2x + 1)^3 \right] = 3(2x + 2)(x^2 + 2x + 1)^2
\]

Do you need more details or have any questions?

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Here are some related questions for further exploration:

1. How would you differentiate \( (2x^2 + 3x + 5)^4 \)?
2. What is the derivative of \( (x^3 - x + 4)^5 \)?
3. Can you apply the product rule to \( x^2(x^2 + 2x + 1)^3 \)?
4. What happens if you expand \( (x^2 + 2x + 1)^3 \) first and then differentiate?
5. How would you integrate \( 3(2x + 2)(x^2 + 2x + 1)^2 \)?

**Tip:** The chain rule is extremely useful for differentiating composite functions. Always identify the outer and inner functions clearly.

Learn how to differentiate the function (x^2 + 2x + 1)^3 using the chain rule with a step-by-step solution. This problem involves differentiation of composite functions and application of the chain rule formula. Suitable for advanced high school students.

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