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The problem in your image asks to find \( H'(3) \), where \( H(x) = 2f(x) + 5g(x) \). The graphs of \( f(x) \) and \( g(x) \) are provided, and we are tasked with calculating \( H'(x) \).

Given that \( H(x) = 2f(x) + 5g(x) \), its derivative is:

\[
H'(x) = 2f'(x) + 5g'(x)
\]

To find \( H'(3) \), we need the slopes of \( f(x) \) and \( g(x) \) at \( x = 3 \), which are the values of \( f'(3) \) and \( g'(3) \). 

Looking at the graph:
- \( f(x) \) (blue line) is linear, so its slope is constant. It appears to have a negative slope, and you can estimate the rise over run to determine \( f'(x) \).
- \( g(x) \) (red line) also has a constant slope, but it's positive. Again, you can estimate the rise over run for \( g'(x) \).

Let's find the slopes:
- \( f'(x) \) appears to have a slope of \( -1 \).
- \( g'(x) \) appears to have a slope of \( 2 \).

Substituting these values into the derivative:

\[
H'(3) = 2(-1) + 5(2) = -2 + 10 = 8
\]

Thus, \( H'(3) = 8 \).

Would you like more details or have any questions?

Here are 5 related questions:
1. How do we determine the derivative of a linear function from its graph?
2. How would \( H(x) \) change if the coefficients of \( f(x) \) and \( g(x) \) were different?
3. What if \( f(x) \) and \( g(x) \) were non-linear functions? How would the process differ?
4. Can you use this approach to calculate \( H'(x) \) at a different point?
5. How does the concept of slope relate to the derivative?

**Tip:** When graphing linear functions, the slope remains constant at all points, making it easier to calculate derivatives directly from the graph.

Let H(x) = 2f(x) + 5g(x), where the graphs of f and g are shown. Find H'(3).

Learn how to calculate the derivative H'(3) for the function H(x) = 2f(x) + 5g(x) using the slopes of linear functions f(x) and g(x) from their graphs. This solution covers basic concepts in calculus, including the sum rule for derivatives. Suitable for students in Grades 11-12.

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