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The given problem asks to determine the slope of the tangent line of the function \( f(x) = x^4 + x^3 \) at \( x = -8 \), using the limit definition of the derivative.

The slope of the tangent line to the curve at a point is given by the derivative of the function at that point. The limit definition of the derivative is:

\[
f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}
\]

In this case, you need to find the slope at \( x = -8 \), so we compute:

\[
f'(-8) = \lim_{h \to 0} \frac{f(-8 + h) - f(-8)}{h}
\]

### Step-by-Step Process:
1. **Find \( f(-8) \):**
   \[
   f(-8) = (-8)^4 + (-8)^3 = 4096 - 512 = 3584
   \]

2. **Find \( f(-8 + h) \):**
   Substitute \( -8 + h \) into the function:
   \[
   f(-8 + h) = (-8 + h)^4 + (-8 + h)^3
   \]

   You do not need to fully expand and simplify this expression, as the problem specifies that simplification is not required. 

3. **Apply the limit:**
   The derivative at \( x = -8 \) is:
   \[
   f'(-8) = \lim_{h \to 0} \frac{(-8 + h)^4 + (-8 + h)^3 - 3584}{h}
   \]

This expression represents the slope of the tangent line at \( x = -8 \). Since the question asks to leave the answer unsimplified, this is the final form.

Would you like further details or explanations on any step? 

Here are some follow-up questions to deepen understanding:

1. What is the geometric meaning of a tangent line to a curve?
2. How does the derivative represent the slope of the tangent line?
3. Why is the limit definition of the derivative important in calculus?
4. How would you compute the full derivative of \( f(x) = x^4 + x^3 \) using differentiation rules?
5. What does the slope of the tangent line tell you about the behavior of the function at \( x = -8 \)?

**Tip**: The limit definition of the derivative is crucial for understanding how rates of change behave at a specific point, providing the foundation for more advanced calculus topics.

Given the function f(x) = x^4 + x^3, determine the slope of the tangent line of f at x = -8 using the limit shown below. You do not have to simplify your answer.

Learn how to determine the slope of the tangent line for the function f(x) = x^4 + x^3 at x = -8 using the limit definition of a derivative. A detailed step-by-step explanation is provided, without needing to simplify the final answer.

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