The given problem asks to identify which of the five graphs represents the derivative $f'$ of a function $f$, given the graph of $f$.

Key Steps to Analyze the Graphs:

1. Identify the Slope of $f$:
   • Where $f$ has a maximum or minimum, the slope of $f$ is zero, so $f' = 0$ at those points.
   • Where $f$ is increasing, the slope is positive, so $f'$ should be positive.
   • Where $f$ is decreasing, the slope is negative, so $f'$ should be negative.

Observations about the Graph of $f$:

• The function $f$ has a maximum at $x = -2$, which means $f'(-2) = 0$.
• The function $f$ has a minimum at $x = 2$, so $f'(2) = 0$.
• The function is increasing on the intervals $(-\infty, -2)$ and $(2, \infty)$, which means $f'$ is positive in these regions.
• The function is decreasing on the interval $(-2, 2)$, meaning $f'$ is negative there.

Compare with the Options:

• Option 1: This graph shows a downward sloping straight line, which does not match the pattern of positive, zero, and negative slopes derived from $f$.
• Option 2: This graph shows the correct pattern: positive before $x = -2$, zero at $x = -2$, negative between $x = -2$ and $x = 2$, zero at $x = 2$, and positive after $x = 2$. This matches the derivative of $f$.
• Option 3: This graph is entirely positive, which does not match the changing slopes of $f$.
• Option 4: Similar to Option 3, this graph shows an incorrect pattern of positive slope throughout.
• Option 5: This graph is negative before and positive after $x = 0$, which does not match the behavior of $f$.

Conclusion:

The correct graph of the derivative $f'$ is Option 2.

Let me know if you would like more details or have any questions! Here are some related questions:

1. What is the graphical relationship between a function and its derivative?
2. How does the second derivative $f''(x)$ relate to the concavity of $f(x)$?
3. What are the critical points of a function, and how are they determined using $f'(x)$?
4. How do inflection points relate to the second derivative $f''(x)$?
5. Can you always find the derivative graphically without knowing the equation of the function?

Tip: When analyzing graphs of functions and their derivatives, always look for where the original function is increasing, decreasing, and has horizontal tangents (maxima or minima). These provide critical clues for sketching the derivative.