https://file.aiquickdraw.com/mathbot/file/1727364019454j3xlytbx.jpg

Let's analyze the image step by step. The table provides values for \( f(x) \), \( g(x) \), and their derivatives \( \frac{d f}{d x} \) and \( \frac{d g}{d x} \) at specific values of \( x \). You are tasked with finding the derivative of \( h(x) \) at certain points based on given functions of \( f(x) \) and \( g(x) \).

The provided formulas for \( h(x) \) are:

1. \( h(x) = -9f(x) + 9g(x) \), and you need to find \( h'(2) \).
2. \( h(x) = 5f(x) - 2g(x) \), and you need to find \( h'(4) \).
3. \( h(x) = 7f(x) - 2g(x) \), and you need to find \( h'(1) \).

### Approach:

The derivative of \( h(x) \) can be computed using the following rule:
\[ h'(x) = -9f'(x) + 9g'(x) \] for the first case,
\[ h'(x) = 5f'(x) - 2g'(x) \] for the second case, 
\[ h'(x) = 7f'(x) - 2g'(x) \] for the third case.

We will then look at the values of \( f'(x) \) and \( g'(x) \) from the table at the respective points of \( x = 2 \), \( x = 4 \), and \( x = 1 \).

### Solving each one:

1. **For \( h'(2) \):**
   \[ h'(2) = -9f'(2) + 9g'(2) \]
   From the table, \( f'(2) = -4 \) and \( g'(2) = -12 \).
   \[
   h'(2) = -9(-4) + 9(-12) = 36 - 108 = -72
   \]

2. **For \( h'(4) \):**
   \[ h'(4) = 5f'(4) - 2g'(4) \]
   From the table, \( f'(4) = 4 \) and \( g'(4) = 0 \).
   \[
   h'(4) = 5(4) - 2(0) = 20
   \]

3. **For \( h'(1) \):**
   \[ h'(1) = 7f'(1) - 2g'(1) \]
   From the table, \( f'(1) = 8 \) and \( g'(1) = -27 \).
   \[
   h'(1) = 7(8) - 2(-27) = 56 + 54 = 110
   \]

### Final Answers:

1. \( h'(2) = -72 \)
2. \( h'(4) = 20 \)
3. \( h'(1) = 110 \)

Would you like further details or clarification on any of the steps?

### Related Questions:
1. What is the chain rule for differentiation, and how is it applied here?
2. How would \( h(x) \) change if it was a product of \( f(x) \) and \( g(x) \)?
3. Can you compute the second derivative \( h''(x) \) for any of these cases?
4. How would you approach this if the functions were not differentiable at certain points?
5. How do you interpret the derivative geometrically?

### Tip:
When applying differentiation rules to composite functions, always verify whether you need to apply additional rules like the product or chain rule!

For the following exercise, assume that f(x) and g(x) are both differentiable functions of x with values as given in the table. Use the table to calculate the following derivatives: h'(2), h'(4), and h'(1) based on different formulas for h(x).

Learn how to calculate derivatives h'(2), h'(4), and h'(1) based on given functions f(x) and g(x) and their derivatives. This example shows step-by-step solutions using linear combinations of functions and differentiation rules.

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