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We are asked to find \( H'(5) \), where \( H(x) = 4f(x) + 2g(x) \), based on the graphs of \( f(x) \) and \( g(x) \).

To do this, we need the following steps:
1. Find the derivatives \( f'(x) \) and \( g'(x) \) at \( x = 5 \) from the graphs, as these derivatives correspond to the slopes of the lines \( y = f(x) \) and \( y = g(x) \) at \( x = 5 \).
2. Then use the derivative rule for linear combinations of functions:
   \[
   H'(x) = 4f'(x) + 2g'(x)
   \]
3. Evaluate this at \( x = 5 \) to find \( H'(5) \).

### Step 1: Find \( f'(5) \) and \( g'(5) \) from the graphs
- From the graph, \( f(x) \) appears to be a straight line. The slope of a straight line is constant, so we can calculate \( f'(x) \) by finding the slope. 
- Similarly, we will calculate the slope for \( g(x) \).

### Calculating Slopes:
1. **For \( f(x) \) (blue line)**: The line passes through points (0, 1) and (5, 3). The slope is given by:
   \[
   f'(x) = \frac{3 - 1}{5 - 0} = \frac{2}{5}
   \]
   So, \( f'(5) = \frac{2}{5} \).

2. **For \( g(x) \) (red line)**: The line passes through points (0, 10) and (5, 0). The slope is given by:
   \[
   g'(x) = \frac{0 - 10}{5 - 0} = \frac{-10}{5} = -2
   \]
   So, \( g'(5) = -2 \).

### Step 2: Use the derivative rule for \( H'(x) \)
\[
H'(x) = 4f'(x) + 2g'(x)
\]
Substitute \( f'(5) = \frac{2}{5} \) and \( g'(5) = -2 \):
\[
H'(5) = 4\left(\frac{2}{5}\right) + 2(-2) = \frac{8}{5} - 4 = \frac{8}{5} - \frac{20}{5} = \frac{-12}{5}
\]

Thus, \( H'(5) = \frac{-12}{5} \).

Would you like further details on this solution or have any other questions?

Here are 5 related questions:
1. How do we find the derivative of a function graphically?
2. What is the significance of linearity in determining derivatives?
3. How would \( H'(x) \) change if the coefficients of \( f(x) \) and \( g(x) \) were different?
4. How can we approximate slopes if the graphs aren't straight lines?
5. How do we apply the chain rule to more complex functions?

**Tip:** For straight lines, the derivative is simply the slope, which makes calculating derivatives visually intuitive.

Let H(x) = 4f(x) + 2g(x), where the graphs of f and g are shown in the figure to the right. Find H'(5).

In this problem, we are asked to find the derivative of H(x) = 4f(x) + 2g(x) at x = 5, using the graphs of f and g. The solution involves calculating the slopes of the lines from the graphs and applying the linear combination rule for derivatives. Suitable for high school students studying calculus.

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