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To find the derivative of the function \( f(x) = x^7 (8x + 4) \), we’ll use the **product rule** of differentiation. The product rule states that if \( f(x) = g(x) \cdot h(x) \), then \( f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x) \).

Let's define:
- \( g(x) = x^7 \)
- \( h(x) = 8x + 4 \)

Then, we’ll find \( g'(x) \) and \( h'(x) \) and apply the product rule.

### Step-by-Step Solution

1. **Differentiate \( g(x) = x^7 \):**
   \[
   g'(x) = 7x^6
   \]

2. **Differentiate \( h(x) = 8x + 4 \):**
   \[
   h'(x) = 8
   \]

3. **Apply the product rule:**
   \[
   f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x)
   \]
   Substitute \( g(x) \), \( g'(x) \), \( h(x) \), and \( h'(x) \):
   \[
   f'(x) = (7x^6)(8x + 4) + (x^7)(8)
   \]

4. **Simplify each term:**

   - For the first term:
     \[
     (7x^6)(8x + 4) = 56x^7 + 28x^6
     \]

   - For the second term:
     \[
     (x^7)(8) = 8x^7
     \]

5. **Combine like terms:**
   \[
   f'(x) = 56x^7 + 28x^6 + 8x^7
   \]
   \[
   = 64x^7 + 28x^6
   \]

### Final Answer
\[
f'(x) = 64x^7 + 28x^6
\]

Would you like further details, or do you have any questions?

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#### Related Questions to Explore:
1. What if \( f(x) = x^7 \cdot (8x^2 + 4) \); how would we find the derivative?
2. How does the product rule differ from the chain rule?
3. Can the solution be verified by expanding \( f(x) \) first and then differentiating?
4. What is the second derivative of \( f(x) = x^7 (8x + 4) \)?
5. How does the power rule apply to functions like \( x^7 \) in differentiation?

#### Tip
Remember to identify and label each component of a function clearly when using the product rule to avoid confusion in complex derivatives.

Find the derivative of the function f(x) = x^7 (8x + 4). (Use symbolic notation and fractions where needed.)

Calculate the derivative of f(x) = x^7 (8x + 4) by applying the product rule of differentiation. This solution provides a step-by-step approach for high school students, covering key calculus concepts such as the product rule and power rule.

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