The graph provided shows the function $f(x)$, and the task is to order the derivatives of $f(x)$ at specific points from least to greatest. The derivative $f'(x)$ represents the slope of the function at each point. Let's analyze the graph step by step:

1. At $x = -4$: The slope is negative as the graph is decreasing steeply.
2. At $x = -3$: The slope is close to zero as the graph flattens near the local minimum.
3. At $x = -2$: The slope is positive because the graph is increasing.
4. At $x = -1$: The slope is negative because the graph is decreasing.
5. At $x = 0$: The slope is positive and increasing steeply.

Ordered from least to greatest:

• $f'(-4)$ is most negative (steepest decrease).
• $f'(-1)$ is negative but not as steep as $f'(-4)$.
• $f'(-3)$ is near zero (local minimum).
• $f'(-2)$ is slightly positive (increasing slowly).
• $f'(0)$ is the largest (steep increase).

So, the order is: $f'(-4) < f'(-1) < f'(-3) < f'(-2) < f'(0)$

Would you like more details or any clarifications? Here are 5 follow-up questions:

1. What does the derivative tell us about the function's behavior at a specific point?
2. How do you determine if the slope is positive or negative from a graph?
3. Can a function have a zero slope and still change direction? Why?
4. What is the significance of points where the derivative equals zero?
5. How would you find the exact values of the derivative at these points?

Tip: When looking at a graph, the steeper the curve, the larger the magnitude of the derivative (either positive or negative).