The problem asks to find the derivative $f'(x)$ of the function $f(x)$, whose graph is provided in the upper-right corner. The graph of $f(x)$ looks like a smooth, bell-shaped curve, which indicates certain key features we can use to determine the graph of $f'(x)$.

Step-by-step Analysis:

1. Critical Points and Slope Behavior:

   • The peak of the curve occurs at $x = 0$. At this point, the slope of the function is zero, meaning $f'(0) = 0$. So, $f'(x)$ should cross the x-axis at $x = 0$.

2. Sign of Slope:

   • To the left of $x = 0$, the function is increasing, which means $f'(x)$ is positive.
   • To the right of $x = 0$, the function is decreasing, so $f'(x)$ should be negative.

3. Concavity and Rate of Change:

   • Before reaching the peak at $x = 0$, the slope gradually increases, so $f'(x)$ starts positive but decreases as it approaches zero.
   • After passing $x = 0$, the slope becomes negative and gradually decreases, implying that $f'(x)$ becomes negative after crossing the x-axis.

Matching the Derivative:

Given the above analysis, we are looking for a graph that:

• Crosses the x-axis at $x = 0$.
• Is positive to the left of $x = 0$.
• Is negative to the right of $x = 0$.
• Shows a smooth transition from positive to negative.

Conclusion:

Graph (B) fits this description. It starts positive, crosses the x-axis at $x = 0$, and then becomes negative, matching the derivative of the bell-shaped curve.

Would you like more details on the reasoning or need further clarification?

Here are 5 relative questions that could expand your understanding:

1. How can we identify critical points and their significance on the graph of a derivative?
2. What does the second derivative tell us about the concavity of the original function?
3. How would the graph of the second derivative look based on the provided function?
4. Why is the slope zero at the maximum or minimum points of a function?
5. What is the relationship between the rate of change of a function and the steepness of its graph?

Tip: When analyzing the derivative graph, remember that it reflects the slope behavior of the original function: positive for increasing slopes, negative for decreasing ones.