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We are given the task of evaluating the derivative of the product \( f(x) \cdot h(x) \) at \( x = -1 \), using the provided table that contains values for \( f(x) \), \( h(x) \), \( f'(x) \), and \( h'(x) \) at \( x = -1 \).

### Formula to Use:
This requires the **product rule** for derivatives, which states:
\[
\frac{d}{dx} [f(x) \cdot h(x)] = f'(x) \cdot h(x) + f(x) \cdot h'(x)
\]

### Step-by-Step Process:

From the table:
- \( f(-1) = -3 \)
- \( h(-1) = -4 \)
- \( f'(-1) = 1 \)
- \( h'(-1) = 3 \)

Now, apply the product rule:
\[
\frac{d}{dx} [f(x) \cdot h(x)] \bigg|_{x=-1} = f'(-1) \cdot h(-1) + f(-1) \cdot h'(-1)
\]

Substituting the values from the table:
\[
\frac{d}{dx} [f(x) \cdot h(x)] \bigg|_{x=-1} = 1 \cdot (-4) + (-3) \cdot 3
\]

Simplifying:
\[
= -4 + (-9) = -4 - 9 = -13
\]

### Final Answer:
\[
\frac{d}{dx} [f(x) \cdot h(x)] \bigg|_{x=-1} = -13
\]

Would you like more details or have any further questions?

Here are 5 related questions to expand on this topic:
1. How does the product rule differ from the chain rule in differentiation?
2. What are some common applications of the product rule in real-world scenarios?
3. How would the result change if either \( f(x) \) or \( h(x) \) were constant?
4. What is the significance of knowing both \( f'(x) \) and \( h'(x) \) in terms of interpreting the behavior of the product?
5. Can the product rule be extended to more than two functions? If so, how?

**Tip:** Always double-check whether you're dealing with a product or a composition of functions to apply the correct differentiation rule.

Evaluate \( \frac{d}{dx} [f(x) \cdot h(x)] \) at \( x = -1 \), given the table of values for \( f(x), h(x), f'(x), h'(x) \).

Learn how to evaluate the derivative of the product of two functions, f(x) and h(x), using the product rule at x = -1. The solution includes step-by-step application of the product rule for differentiation, referencing key values of f(x), h(x), and their derivatives.

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