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We are given that \( G(x) = 2f(x) - g(x) \) and asked to find \( G'(5) \). To solve this, we'll use the derivative:

\[
G'(x) = 2f'(x) - g'(x)
\]

We need to find \( f'(5) \) and \( g'(5) \) from the graphs of \( f(x) \) and \( g(x) \) provided.

1. **Slope of \( f(x) \) at \( x = 5 \):** Looking at the graph of \( f(x) \) (the blue line), the segment between \( x = 4 \) and \( x = 6 \) is a straight line. The slope of a line segment is the change in \( y \) over the change in \( x \). From the graph, it appears that between \( x = 4 \) and \( x = 6 \), \( f(x) \) decreases from 4 to 0, so:
   \[
   f'(5) = \frac{0 - 4}{6 - 4} = \frac{-4}{2} = -2
   \]

2. **Slope of \( g(x) \) at \( x = 5 \):** For the red line representing \( g(x) \), the segment between \( x = 4 \) and \( x = 6 \) is also a straight line. Between \( x = 4 \) and \( x = 6 \), \( g(x) \) increases from 0 to 4, so:
   \[
   g'(5) = \frac{4 - 0}{6 - 4} = \frac{4}{2} = 2
   \]

Now that we have \( f'(5) = -2 \) and \( g'(5) = 2 \), we can compute \( G'(5) \):

\[
G'(5) = 2f'(5) - g'(5) = 2(-2) - 2 = -4 - 2 = -6
\]

Thus, \( G'(5) = -6 \).

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Would you like more details on any of the steps, or have any other questions?

Here are some related questions to explore further:
1. How would the derivative change if the function was \( G(x) = f(x) - 2g(x) \)?
2. How do you calculate the slope for nonlinear functions?
3. What happens if the functions \( f(x) \) and \( g(x) \) are not linear at \( x = 5 \)?
4. How does the chain rule apply when differentiating composed functions like \( G(x) = f(g(x)) \)?
5. Can you find \( G'(x) \) for other points, like \( G'(3) \)?

**Tip:** When working with piecewise linear functions, calculating the slope is as simple as finding the rise over run between the endpoints of the linear segments.

Let G = 2f - g, where the graphs of f and g are shown. Find G'(5).

Solve the derivative of G(x) = 2f(x) - g(x) at x = 5 using the graphs of f(x) and g(x). This problem involves calculating the slopes of linear functions from their graphs and applying basic derivative rules. Suitable for students in Grades 10-12.

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