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To sketch the graph of the derivative \( f' \) from the given graph of \( f \), you need to analyze the following key points and behaviors of \( f \):

### 1. **Critical Points (Where the Slope is Zero)**
   - Wherever the function \( f(x) \) has local maxima or minima, the derivative \( f'(x) \) will be zero because the slope of \( f(x) \) is zero at these points.
   - In the graph, maxima occur around \( x = -6 \), \( x = 0 \), and \( x = 6 \). Minima occur around \( x = -3 \) and \( x = 3 \).

### 2. **Slope Behavior**
   - Where \( f(x) \) is increasing (positive slope), the derivative \( f'(x) \) will be positive.
   - Where \( f(x) \) is decreasing (negative slope), the derivative \( f'(x) \) will be negative.

### 3. **Inflection Points (Where the Slope is Changing)**
   - The points where the concavity of the function changes indicate inflection points. At these points, the derivative will cross the x-axis and change signs.

### Key Observations for \( f'(x) \):
   - \( f(x) \) is increasing between \( x = -9 \) and \( x = -6 \), so \( f'(x) \) should be positive.
   - At \( x = -6 \), \( f(x) \) has a local maximum, meaning \( f'(x) = 0 \) at this point.
   - From \( x = -6 \) to \( x = -3 \), \( f(x) \) is decreasing, so \( f'(x) \) is negative.
   - At \( x = -3 \), \( f(x) \) has a local minimum, meaning \( f'(x) = 0 \).
   - This behavior repeats periodically with the same pattern for the other critical points.

By following these steps, you can sketch the graph of \( f'(x) \), which should resemble a sinusoidal curve that crosses the x-axis at the maxima and minima of \( f(x) \) and has positive and negative regions corresponding to the increasing and decreasing intervals of \( f(x) \).

Let me know if you would like further details or have any questions!

### Expanding Questions:
1. What would the graph of the second derivative \( f''(x) \) look like based on the behavior of \( f(x) \)?
2. How can you identify inflection points on the graph of \( f(x) \) and interpret them in terms of \( f'(x) \)?
3. What happens to the derivative \( f'(x) \) at points of concavity change in \( f(x) \)?
4. How can we approximate the steepness of \( f(x) \) to sketch the magnitude of \( f'(x) \)?
5. What are the implications of symmetry in the graph of \( f(x) \) for the derivative \( f'(x) \)?

### Tip:
Remember, the derivative represents the rate of change or slope at each point on the graph of \( f(x) \). A positive slope means the graph of \( f(x) \) is rising, and a negative slope means it is falling.

The graph of a function f is shown below. Sketch the graph of the derivative f'.

Analyze a graph of a function f(x) and learn how to sketch the derivative f'(x) by identifying key points such as maxima, minima, and intervals of increase or decrease. This problem covers important calculus concepts suitable for students in advanced high school or college-level calculus.

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