To construct the graph of $f'$, the derivative of $f$, from the graph of $f$, follow these steps:

Key observations about $f$ in the graph:

1. Critical points (where $f'(x) = 0$) occur where $f$ has horizontal tangents (flat slopes).
2. Concavity changes (inflection points) occur where $f''(x) = 0$ and the concavity switches between up and down.
3. The slope of $f$ tells us the value of $f'$ at any point.

Step-by-step observations:

1. Identify where $f$ has horizontal tangents:

   • From the graph, these occur approximately at $x = -6$, $x = 2$, and $x = 7$. At these points, $f'(x) = 0$, so $f'$ will cross the $x$-axis.

2. Determine where $f$ is increasing or decreasing:

   • $f$ is decreasing for $x < -6$, so $f'(x) < 0$ in this region.
   • $f$ is increasing between $x = -6$ and $x = 2$, so $f'(x) > 0$ here.
   • $f$ decreases again between $x = 2$ and $x = 7$, so $f'(x) < 0$ here.
   • $f$ increases for $x > 7$, so $f'(x) > 0$ in this region.

3. Analyze concavity changes (green dots, $f''(x) = 0$):

   • The inflection points at $x \approx -3$ and $x \approx 5$ indicate that $f'$ has critical points (local maxima or minima) at these positions.

Sketching $f'$:

• At $x = -6$, $x = 2$, and $x = 7$, the graph of $f'$ crosses the $x$-axis.
• Between these points:
  • $f'$ starts negative (slopes of $f$ are decreasing).
  • Peaks or valleys in $f'$ occur at the inflection points of $f$.
• Ensure $f'$ reflects the changing slopes of $f$:
  • Decreasing slopes mean $f'$ is negative.
  • Increasing slopes mean $f'$ is positive.

Let me know if you need further clarification or if you'd like me to summarize this into steps for sketching the graph!

Follow-up Questions:

1. What is the significance of $f'(x) = 0$ for the graph of $f$?
2. How do inflection points of $f$ affect the graph of $f'$?
3. How can you determine $f''(x)$ from the graph of $f$?
4. What does the sign of $f'(x)$ tell you about $f$?
5. How would you sketch $f''$ based on the graph of $f'$?

Tip: Remember that $f'$ represents the slopes of $f$, so interpreting changes in slopes is key to constructing $f'$.