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We are given the function \( f(x) = 8x^4 \arctan(7x^2) \) and asked to find \( f'(x) \), the derivative.

To differentiate this, we will need to apply both the **product rule** and the **chain rule**. The product rule states that if \( f(x) = u(x) \cdot v(x) \), then:

\[
f'(x) = u'(x) v(x) + u(x) v'(x)
\]

Here:
- \( u(x) = 8x^4 \)
- \( v(x) = \arctan(7x^2) \)

### Step 1: Differentiate \( u(x) = 8x^4 \)
\[
u'(x) = 32x^3
\]

### Step 2: Differentiate \( v(x) = \arctan(7x^2) \)
For this, we use the chain rule. The derivative of \( \arctan(g(x)) \) is:

\[
\frac{d}{dx} \arctan(g(x)) = \frac{g'(x)}{1 + g(x)^2}
\]

Here, \( g(x) = 7x^2 \), so:
- \( g'(x) = 14x \)

Thus, the derivative of \( v(x) = \arctan(7x^2) \) is:

\[
v'(x) = \frac{14x}{1 + (7x^2)^2} = \frac{14x}{1 + 49x^4}
\]

### Step 3: Apply the product rule
Now, apply the product rule to \( f(x) = 8x^4 \arctan(7x^2) \):

\[
f'(x) = u'(x) v(x) + u(x) v'(x)
\]
Substituting the expressions for \( u'(x) \), \( v(x) \), \( u(x) \), and \( v'(x) \):

\[
f'(x) = 32x^3 \arctan(7x^2) + 8x^4 \cdot \frac{14x}{1 + 49x^4}
\]

Simplifying the second term:

\[
f'(x) = 32x^3 \arctan(7x^2) + \frac{112x^5}{1 + 49x^4}
\]

Thus, the derivative of \( f(x) \) is:

\[
f'(x) = 32x^3 \arctan(7x^2) + \frac{112x^5}{1 + 49x^4}
\]

Let me know if you need any further clarification on the steps or have additional questions! Here are 5 related questions you might find useful:

1. How would you apply the product rule to more complicated functions?
2. What happens if you differentiate \( \arctan(x) \) directly without a chain rule?
3. How would the derivative change if the function inside \( \arctan \) were different?
4. Can the chain rule be applied to any composition of functions?
5. What is the derivative of other inverse trigonometric functions?

**Tip:** When dealing with trigonometric functions, it's helpful to memorize their derivatives and know when to apply the chain rule effectively.

Learn how to differentiate the function f(x) = 8x^4 arctan(7x^2) by applying the product rule and chain rule. This problem is ideal for students in Grade 12 or college-level calculus.

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