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To find the derivative of \( f(x) = (x + x^7)(-x - 4) \) using the product rule, let's define each part of the product and proceed step-by-step.

### Step 1: Define the components
Let:
\[
u(x) = x + x^7
\]
\[
v(x) = -x - 4
\]

### Step 2: Find \( u'(x) \) and \( v'(x) \)
1. **Differentiate \( u(x) \):**
   \[
   u(x) = x + x^7
   \]
   \[
   u'(x) = 1 + 7x^6
   \]

2. **Differentiate \( v(x) \):**
   \[
   v(x) = -x - 4
   \]
   \[
   v'(x) = -1
   \]

### Step 3: Apply the Product Rule
The product rule states:
\[
f'(x) = u'(x)v(x) + u(x)v'(x)
\]

Now substitute \( u(x) \), \( u'(x) \), \( v(x) \), and \( v'(x) \) into this formula:
\[
f'(x) = (1 + 7x^6)(-x - 4) + (x + x^7)(-1)
\]

### Step 4: Simplify \( f'(x) \)
Expand each term:

1. **First term:**
   \[
   (1 + 7x^6)(-x - 4) = -x - 4 - 7x^7 - 28x^6
   \]

2. **Second term:**
   \[
   (x + x^7)(-1) = -x - x^7
   \]

Now, combine these:
\[
f'(x) = (-x - 4 - 7x^7 - 28x^6) + (-x - x^7)
\]

Combine like terms:
\[
f'(x) = -8x^7 - 28x^6 - 2x - 4
\]

### Final Answer:
\[
f'(x) = -8x^7 - 28x^6 - 2x - 4
\]

Let me know if you have any questions or if you'd like further clarification on any part of the process!

### Related Questions to Explore:
1. How does the product rule work with three functions multiplied together?
2. What is the chain rule, and how does it differ from the product rule?
3. How would you approach finding the derivative if the expression were a quotient?
4. Why is it important to simplify derivatives after using the product rule?
5. Can the product rule be applied iteratively for multiple products? 

**Tip:** Always simplify expressions where possible after applying the product rule, as it makes further calculations easier and reduces the chances of errors.

Find the derivative of f using the product rule.

f(x) = (x + x^7)(-x - 4)

Learn how to find the derivative of f(x) = (x + x^7)(-x - 4) using the product rule. This solution provides a step-by-step approach, breaking down the application of the product rule and simplifying the derivative.

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