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To solve the question of which graph corresponds to \( f'(x) \), the derivative of \( f(x) \), we need to analyze the behavior of the given function, \( f(x) \), to infer information about its derivative.

### Key Observations from the graph of \( f(x) \):

1. **Slope analysis (derivative):**
   - Where the slope of the graph of \( f(x) \) is **positive**, the derivative \( f'(x) \) will be **positive**.
   - Where the slope is **negative**, \( f'(x) \) will be **negative**.
   - Where the slope is **zero** (flat), \( f'(x) = 0 \). These points are critical points of \( f(x) \) and correspond to **zeros** on the graph of \( f'(x) \).

2. **Critical points (slope = 0):**
   - The graph of \( f(x) \) is flat at approximately \( x = -10 \), \( x = 0 \), and perhaps at a point around \( x = 5 \). These points indicate that \( f'(x) = 0 \) at those values of \( x \), meaning the derivative will cross the x-axis at these points.

3. **Regions of increasing and decreasing:**
   - From \( x = -15 \) to about \( x = -10 \), the graph is steeply decreasing, so \( f'(x) \) should be **negative**.
   - From \( x = -10 \) to \( x = 0 \), the graph is increasing, so \( f'(x) \) is **positive**.
   - From \( x = 0 \) to about \( x = 5 \), the graph is slightly decreasing, indicating a **negative** derivative in this region.
   - After \( x = 5 \), the function is increasing again, implying that \( f'(x) \) will be **positive**.

By analyzing the slope behavior of the original function \( f(x) \), we can predict the shape of \( f'(x) \), which should have:
- Zeros at \( x = -10 \), \( x = 0 \), and \( x = 5 \).
- Positive regions where \( f(x) \) is increasing.
- Negative regions where \( f(x) \) is decreasing.

Would you like to explore more details about how derivative graphs are shaped, or would you like to upload potential options to confirm the graph of \( f'(x) \)? Here are some related questions for further exploration:

1. How does the second derivative help in determining concavity in the graph of \( f(x) \)?
2. What happens to the derivative graph at points of inflection?
3. How would the derivative graph change if the original function had more pronounced peaks or valleys?
4. What role do critical points play in analyzing the behavior of \( f'(x) \)?
5. How do we analyze the behavior of \( f'(x) \) at the boundaries of the graph?

**Tip**: Always analyze the critical points where the slope is zero to determine key features of the derivative graph.

Which one of the following graphs corresponds to f'(x), the first derivative of f(x)?

Analyze the graph of a function f(x) to determine the corresponding graph of its first derivative, f'(x). This problem involves critical points, slope analysis, and regions of increasing and decreasing behavior. Suitable for students in Grades 11-12 or early college calculus.

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