https://file.aiquickdraw.com/mathbot/file/17261058899637tfzsmi9.jpg

To find \( H'(1) \) given that \( H(x) = 4f(x) + g(x) \), we need to differentiate \( H(x) \):

\[
H'(x) = 4f'(x) + g'(x)
\]

We then substitute \( x = 1 \):

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

Now, we need to find the derivatives \( f'(1) \) and \( g'(1) \) using the graphs of \( f(x) \) and \( g(x) \).

### Step 1: Find \( f'(1) \)

From the graph, the function \( f(x) \) is a straight line (blue line), so its slope is constant. The slope can be found as:

\[
f'(x) = \frac{\Delta y}{\Delta x} = \frac{4 - 0}{4 - 0} = 1
\]

Thus, \( f'(1) = 1 \).

### Step 2: Find \( g'(1) \)

Similarly, for \( g(x) \) (red line), its slope is also constant:

\[
g'(x) = \frac{\Delta y}{\Delta x} = \frac{0 - 10}{4 - 0} = -\frac{10}{4} = -2.5
\]

Thus, \( g'(1) = -2.5 \).

### Step 3: Substitute into the derivative of \( H(x) \)

\[
H'(1) = 4 \cdot 1 + (-2.5) = 4 - 2.5 = 1.5
\]

### Final Answer:

\[
H'(1) = 1.5
\]

Would you like further details or have any questions? Here are some related questions to explore:

1. How do you differentiate a linear combination of functions?
2. How do you find the slope of a line from a graph?
3. What is the significance of the derivative in understanding the rate of change?
4. Can we find the second derivative \( H''(x) \) using the given functions?
5. What are other real-life applications of derivatives in problem-solving?

**Tip:** When dealing with graphs of functions, always look for consistent patterns in slope to simplify derivative calculations!

Let H(x) = 4f(x) + g(x), where the graphs of f and g are shown in the figure to the right. Find H'(1).

Explore how to find the derivative H'(1) for the function H(x) = 4f(x) + g(x), using the slopes derived from the graph of each function. This problem illustrates the application of the linearity of derivatives and the constant multiplication rule in calculus.

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