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We are asked to find the derivative \( F'(4) \), where \( F(x) = f(x) + g(x) \). From the graph, we can see that we have two functions \( f(x) \) (blue) and \( g(x) \) (red), and we are tasked with calculating the sum of their derivatives at \( x = 4 \).

### To solve this:
1. **Differentiate** \( F(x) = f(x) + g(x) \), which gives:
   \[
   F'(x) = f'(x) + g'(x)
   \]
2. **Find the slopes** of the tangent lines to the graphs of \( f(x) \) and \( g(x) \) at \( x = 4 \), which represent their derivatives \( f'(4) \) and \( g'(4) \).

### From the graph:
- For \( f(x) \) (blue line):
  - At \( x = 4 \), \( f(x) \) is a straight line with a negative slope. Looking at two clear points on this line (e.g., \( (2, 4) \) and \( (4, 0) \)), we can calculate the slope as:
    \[
    f'(4) = \frac{0 - 4}{4 - 2} = \frac{-4}{2} = -2
    \]
- For \( g(x) \) (red line):
  - At \( x = 4 \), \( g(x) \) is a straight line with a positive slope. Observing points on the line (e.g., \( (4, 4) \) and \( (6, 0) \)), the slope is:
    \[
    g'(4) = \frac{0 - 4}{6 - 4} = \frac{-4}{2} = -2
    \]

### Therefore:
\[
F'(4) = f'(4) + g'(4) = -2 + (-2) = -4
\]

The value of \( F'(4) \) is \(\boxed{-4}\).

Would you like further details on this process or additional explanations?

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Here are 5 follow-up questions to explore related concepts:
1. How do you interpret the derivative of a function at a point graphically?
2. How does the slope of a line between two points help in determining the derivative?
3. What happens if one of the functions is not linear—how do we find the derivative at a point then?
4. Can you explain the physical meaning of adding derivatives of two functions?
5. How would you calculate the second derivative of \( F(x) = f(x) + g(x) \) at \( x = 4 \)?

**Tip**: When calculating slopes from graphs, be sure to choose points that are clearly marked to avoid errors.

Let F = f + g, where the graphs of f and g are shown. Find F'(4).

Learn how to compute the derivative F'(4) for the function F(x) = f(x) + g(x) by analyzing the slopes of the graphs of f and g. This involves key calculus concepts such as differentiation and using slopes of lines to find derivatives. Suitable for high school students studying calculus.

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