To find the derivatives $f'(1)$ and $f'(6)$ based on the graph of the function, we'll follow these steps:

Step 1: Understanding the Graph

You mentioned that the graph consists of two connected line segments:

1. The first line segment goes from $(1, 0)$ to $(4, 6)$.
2. The second line segment goes from $(4, 6)$ to $(8, 8)$.

Since the function consists of line segments, its derivative will be the slope of the line at the given points.

Step 2: Finding the Slope of the Line Segments

The slope of a line segment can be calculated using the formula for the slope of a line: $m = \frac{y_2 - y_1}{x_2 - x_1}$ where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points on the line.

For $f'(1)$:

Since the point $(1, 0)$ lies on the first line segment, and the line segment goes from $(1, 0)$ to $(4, 6)$, we can calculate the slope of this line: $f'(1) = \frac{6 - 0}{4 - 1} = \frac{6}{3} = 2$ Thus, $f'(1) = 2$.

For $f'(6)$:

The point $(6, y)$ lies on the second line segment, which goes from $(4, 6)$ to $(8, 8)$. We calculate the slope of this line: $f'(6) = \frac{8 - 6}{8 - 4} = \frac{2}{4} = 0.5$ Thus, $f'(6) = 0.5$.

Final Answers:

• $f'(1) = 2$
• $f'(6) = 0.5$

Would you like further details on how the slope affects the graph, or do you have any other questions? Here are some related questions to expand on the topic:

1. How does the concept of the slope of a line relate to the derivative of a function?
2. What happens to the slope of the function at points where the function is not differentiable?
3. How can you determine whether a function is increasing or decreasing from its derivative?
4. Can the derivative of a function be negative? If so, what does it signify about the graph?
5. How would you calculate the derivative at a point where the graph has a corner or cusp?

Tip: When working with piecewise linear graphs, you can calculate the derivative by simply finding the slope of each segment of the graph.