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The problem asks to find \( H'(3) \), where \( H(x) = 5f(x) + g(x) \), and the graphs of \( f(x) \) and \( g(x) \) are provided.

We need to compute the derivative of \( H(x) \), which is:

\[
H'(x) = 5f'(x) + g'(x)
\]

To find \( H'(3) \), we must evaluate both \( f'(3) \) and \( g'(3) \). These derivatives represent the slopes of the tangent lines to the graphs of \( f(x) \) and \( g(x) \) at \( x = 3 \).

### Step 1: Determine \( f'(x) \) and \( g'(x) \) from the graphs

- **\( f(x) \)** appears as a linear function with a positive slope.
- **\( g(x) \)** appears as a linear function with a negative slope.

Since both functions are linear, their slopes (derivatives) are constant. We can calculate the slopes from the graphs.

#### For \( f(x) \):
Looking at the graph, \( f(x) \) rises from \( y = 0 \) at \( x = 0 \) to \( y = 6 \) at \( x = 6 \). Thus, the slope \( f'(x) \) is:

\[
f'(x) = \frac{6 - 0}{6 - 0} = 1
\]

#### For \( g(x) \):
\( g(x) \) falls from \( y = 8 \) at \( x = 0 \) to \( y = 0 \) at \( x = 8 \). Thus, the slope \( g'(x) \) is:

\[
g'(x) = \frac{0 - 8}{8 - 0} = -1
\]

### Step 2: Find \( H'(3) \)
Since the derivatives are constant for these linear functions, we have:

\[
f'(3) = 1 \quad \text{and} \quad g'(3) = -1
\]

Now, substitute into the expression for \( H'(x) \):

\[
H'(3) = 5f'(3) + g'(3) = 5(1) + (-1) = 5 - 1 = 4
\]

Thus, \( H'(3) = 4 \).

Would you like further details on any step?

### Related Questions:
1. How do you find the slope of a line using points on a graph?
2. Why is the derivative of a linear function constant?
3. How does the chain rule apply in more complex cases involving product or quotient of functions?
4. Can you generalize the result if \( f(x) \) and \( g(x) \) were nonlinear functions?
5. What is the geometric interpretation of the derivative of a function at a given point?

### Tip:
For linear functions, the slope is constant and can be determined by selecting any two points on the line!

Let H(x) = 5f(x) + g(x), where the graphs of f and g are shown in the figure. Find H'(3).

Calculate H'(3) for the function H(x) = 5f(x) + g(x) using the derivatives of linear functions f and g. This problem involves finding the slopes of tangent lines at x = 3 from the graphs provided. Suitable for students in Grades 10-12.

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