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We are tasked with finding \( G'(6) \), given that \( G(x) = 7f(x) - g(x) \), where the graphs of \( f(x) \) and \( g(x) \) are provided. We need to evaluate the derivative of \( G \) at \( x = 6 \).

### Steps:

1. **Find \( G'(x) \):**
   \[
   G(x) = 7f(x) - g(x)
   \]
   Taking the derivative of both sides:
   \[
   G'(x) = 7f'(x) - g'(x)
   \]
   So, we need to compute \( f'(6) \) and \( g'(6) \) from the graphs.

2. **Determine \( f'(6) \) and \( g'(6) \):**
   The derivative at a specific point is the slope of the tangent line at that point.

   - **For \( f(x) \) at \( x = 6 \):**  
     From the graph, \( f(x) \) appears to be a straight line with a positive slope after \( x = 4 \). We can calculate the slope between points \( (4, 4) \) and \( (8, 8) \), giving:
     \[
     f'(6) = \frac{8 - 4}{8 - 4} = 1
     \]

   - **For \( g(x) \) at \( x = 6 \):**  
     From the graph, \( g(x) \) appears to have a negative slope between \( x = 4 \) and \( x = 8 \). The slope between points \( (4, 8) \) and \( (8, 0) \) is:
     \[
     g'(6) = \frac{0 - 8}{8 - 4} = \frac{-8}{4} = -2
     \]

3. **Evaluate \( G'(6) \):**
   Using the values for \( f'(6) \) and \( g'(6) \):
   \[
   G'(6) = 7(1) - (-2) = 7 + 2 = 9
   \]

Thus, \( G'(6) = 9 \).

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Would you like further details on the steps, or do you have any questions? Here are 5 related questions you might find useful:

1. How do you calculate the derivative from a graph in general?
2. What is the geometric interpretation of the derivative at a point?
3. How would the solution change if the functions were non-linear?
4. What is the physical meaning of multiplying a derivative by a constant, like 7 in this problem?
5. Can you explain the relationship between the slopes of different functions and their derivatives?

**Tip:** Always check the slope of the graph carefully when calculating derivatives from visual data.

Let G = 7f - g, where the graphs of f and g are shown in the figure. Find G'(6).

In this problem, you are tasked with finding the derivative G'(6) for the function G(x) = 7f(x) - g(x) by interpreting the slopes of the graphs of f(x) and g(x) at x = 6. The solution involves using basic differentiation rules and calculating slopes from graph points. Suitable for students in Grades 10-12.

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