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Let's go through the questions step-by-step. Here's what each part asks:

### Problem 6 (Table-based Questions):

**Using the table provided**:
- The table includes values for \( f(x) \), \( f'(x) \), \( g(x) \), and \( g'(x) \) at different \( x \) values.

| x  | f(x) | f'(x) | g(x) | g'(x) |
|----|------|-------|------|-------|
| -2 |  -6  |   9   | -10  |  16   |
|  1 |   5  |   5   |  -3  |  -2   |

We will use the table to answer the following:

1. **Find \( h'(1) \) if \( h(x) = 5 - f(x) \).**
   - Given: \( h(x) = 5 - f(x) \).
   - To find \( h'(x) \), take the derivative: 
     \[
     h'(x) = -f'(x)
     \]
     So, \( h'(1) = -f'(1) = -5 \) (from the table).
   - **Answer: \( h'(1) = -5 \)**.

2. **Find \( k'(-2) \) if \( k(x) = -\frac{1}{2}g(x) \).**
   - Given: \( k(x) = -\frac{1}{2}g(x) \).
   - To find \( k'(x) \), take the derivative:
     \[
     k'(x) = -\frac{1}{2}g'(x)
     \]
     So, \( k'(-2) = -\frac{1}{2} \times 16 = -8 \) (using \( g'(-2) = 16 \) from the table).
   - **Answer: \( k'(-2) = -8 \)**.

3. **Find \( L'(-2) \) if \( L(x) = 2f(x) + 3g(x) \).**
   - Given: \( L(x) = 2f(x) + 3g(x) \).
   - To find \( L'(x) \), take the derivative:
     \[
     L'(x) = 2f'(x) + 3g'(x)
     \]
     So, \( L'(-2) = 2 \times 9 + 3 \times 16 = 18 + 48 = 66 \) (using \( f'(-2) = 9 \) and \( g'(-2) = 16 \) from the table).
   - **Answer: \( L'(-2) = 66 \)**.

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### Would you like me to work through the remaining derivative rules (problems 4–7) or explain any part of this solution further?

### Related Questions:
1. How do you apply the chain rule when finding the derivative of composite functions?
2. What are the common derivative rules for exponential and logarithmic functions?
3. How does the product rule apply when differentiating two functions multiplied together?
4. What is the difference between implicit and explicit differentiation?
5. Can you explain how to apply the quotient rule for rational functions?

### Tip:
For functions that are constants subtracted or added to other functions, the derivative of the constant will always be zero, simplifying the differentiation process.

Using the table provided, compute the following:
1. Find h'(1) if h(x) = 5 - f(x).
2. Find k'(-2) if k(x) = -1/2g(x).
3. Find L'(-2) if L(x) = 2f(x) + 3g(x).

Learn how to calculate derivatives using the sum and constant multiple rules from a table of function values. This problem walks through key concepts of derivatives including the derivative of sums, constant multiples, and differentiating using table data. Suitable for first-year calculus students.

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