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.