To differentiate the function $f(x) = (x^3 + 2x^2 + x)^4$, we'll use the chain rule.

Step 1: Identify the outer and inner functions

• The outer function is $g(u) = u^4$, where $u = x^3 + 2x^2 + x$.
• The inner function is $h(x) = x^3 + 2x^2 + x$.

Step 2: Differentiate the outer function

• The derivative of the outer function $g(u) = u^4$ with respect to $u$ is $g'(u) = 4u^3$.

Step 3: Differentiate the inner function

• The derivative of the inner function $h(x) = x^3 + 2x^2 + x$ with respect to $x$ is:

$h'(x) = \frac{d}{dx} \left( x^3 + 2x^2 + x \right) = 3x^2 + 4x + 1$

Step 4: Apply the chain rule

Using the chain rule:

$\frac{d}{dx} \left[ (x^3 + 2x^2 + x)^4 \right] = 4(x^3 + 2x^2 + x)^3 \cdot (3x^2 + 4x + 1)$

Final Answer:

$\frac{d}{dx} \left[ (x^3 + 2x^2 + x)^4 \right] = 4(3x^2 + 4x + 1)(x^3 + 2x^2 + x)^3$

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 $(x^4 + x^3 + x^2 + x)^5$?
2. What is the derivative of $(x^2 + x + 1)^6$?
3. Can you apply the chain rule to $(2x^2 + 3x + 5)^3$?
4. What happens if you expand $(x^3 + 2x^2 + x)^4$ first and then differentiate?
5. How would you integrate $4(3x^2 + 4x + 1)(x^3 + 2x^2 + x)^3$?

Tip: When applying the chain rule, ensure you clearly identify both the outer and inner functions to correctly apply the derivative.