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We are tasked with finding the derivative \( f'(x) \) of the function \( f(x) = 4x^3 \arctan(4x^3) \).

### Step 1: Recognize the product rule
The function is a product of two parts:
1. \( 4x^3 \)
2. \( \arctan(4x^3) \)

To differentiate a product, we apply the product rule:

\[
f'(x) = \frac{d}{dx}[4x^3] \cdot \arctan(4x^3) + 4x^3 \cdot \frac{d}{dx}[\arctan(4x^3)]
\]

### Step 2: Differentiate each part
1. **Differentiate \( 4x^3 \)**:
\[
\frac{d}{dx}[4x^3] = 12x^2
\]

2. **Differentiate \( \arctan(4x^3) \)** using the chain rule:
The derivative of \( \arctan(u) \) is \( \frac{1}{1+u^2} \), and here \( u = 4x^3 \), so we apply the chain rule:

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

The derivative of \( 4x^3 \) is \( 12x^2 \), so:

\[
\frac{d}{dx}[\arctan(4x^3)] = \frac{12x^2}{1 + 16x^6}
\]

### Step 3: Apply the product rule
Now, we substitute these derivatives back into the product rule expression:

\[
f'(x) = 12x^2 \cdot \arctan(4x^3) + 4x^3 \cdot \frac{12x^2}{1 + 16x^6}
\]

Simplify the second term:

\[
f'(x) = 12x^2 \arctan(4x^3) + \frac{48x^5}{1 + 16x^6}
\]

Thus, the derivative is:

\[
f'(x) = 12x^2 \arctan(4x^3) + \frac{48x^5}{1 + 16x^6}
\]

This is the final answer for \( f'(x) \).

Would you like further details or have any questions?

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Here are 5 related questions to consider:
1. How is the product rule applied in differentiation?
2. Can you explain the chain rule and how it was used in this problem?
3. How does the derivative of \( \arctan(u) \) work in general?
4. What are some other examples of functions that require the product rule to differentiate?
5. What happens if you differentiate the result obtained for \( f'(x) \) again?

**Tip:** When applying the product rule, always make sure to fully differentiate both functions and don't forget about the chain rule when necessary.

Learn how to differentiate the function f(x) = 4x^3 arctan(4x^3) by applying the product rule and chain rule. This detailed step-by-step solution also covers the differentiation of trigonometric functions. Suitable for high school students in Grades 11-12.

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