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The given problem asks you to sketch the graph of \( f'(x) \), the derivative of \( f(x) \), based on the graph of \( f(x) \) provided.

### Key points to observe:
1. **Slope behavior**: The derivative \( f'(x) \) reflects the slope of \( f(x) \). When \( f(x) \) is increasing, \( f'(x) \) is positive, and when \( f(x) \) is decreasing, \( f'(x) \) is negative.
2. **Critical points**: The points where the slope of \( f(x) \) is zero (flat regions or peaks/valleys) will correspond to where \( f'(x) = 0 \).

### Analyzing \( f(x) \):
- From \( x = -4 \) to \( x = -2 \), \( f(x) \) is increasing, so \( f'(x) > 0 \).
- At \( x = -2 \), there is a peak, so the slope is zero, meaning \( f'(x) = 0 \).
- From \( x = -2 \) to \( x = 1 \), \( f(x) \) is decreasing, so \( f'(x) < 0 \).
- At \( x = 1 \), there is a valley, so the slope is zero, meaning \( f'(x) = 0 \).
- From \( x = 1 \) to \( x = 3 \), \( f(x) \) is increasing again, so \( f'(x) > 0 \).
- From \( x = 3 \) to \( x = 4 \), \( f(x) \) is flat, so \( f'(x) = 0 \).

### Steps to sketch \( f'(x) \):
1. Plot points where the slope of \( f(x) \) is zero: \( f'(x) = 0 \) at \( x = -2 \), \( x = 1 \), and \( x = 3 \).
2. Identify regions of increasing and decreasing slopes:
   - \( f'(x) > 0 \) for \( -4 < x < -2 \) and \( 1 < x < 3 \).
   - \( f'(x) < 0 \) for \( -2 < x < 1 \).
   - \( f'(x) = 0 \) at \( x = 3 \) because the slope is flat after that.

The resulting graph will have positive values for \( f'(x) \) on intervals where \( f(x) \) increases and negative values where \( f(x) \) decreases, with zero values at the critical points.

Would you like a more detailed step-by-step graphing of \( f'(x) \) or any specific clarifications? 

Here are 5 related questions:
1. How does the graph of \( f'(x) \) relate to the concavity of \( f(x) \)?
2. What happens to \( f'(x) \) if \( f(x) \) is a constant function?
3. How would the graph of \( f''(x) \) behave based on the graph of \( f(x) \)?
4. Why is the derivative zero at local maximum and minimum points?
5. How can you interpret the slope of a graph in terms of real-world problems?

**Tip**: When analyzing the graph of \( f(x) \), always look for points where the slope changes direction to identify critical points for \( f'(x) \).

Given the following graph of a function f(x), sketch the graph of f'(x).

This problem asks students to sketch the graph of the derivative f'(x) using the given graph of f(x). It covers key concepts such as identifying increasing/decreasing intervals and finding critical points. Suitable for high school students (Grades 10-12).

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