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.