1. **Problem Statement:** Given the graph of a function $f(x)$, determine the graph of its derivative $f'(x)$. The derivative graph represents the slope of $f(x)$ at each point. 2. **Key Concept:** The derivative $f'(x)$ at any point is the slope of the tangent line to $f(x)$ at that point. 3. **Analyzing the graph of $f(x)$:** - From $x=-6$ to about $x=-2$, $f(x)$ is increasing steeply, so $f'(x)$ is positive and large. - Near $x=-1$, $f(x)$ has a local maximum, so $f'(x) = 0$ there. - Between $x=-1$ and $x=2$, $f(x)$ decreases to a local minimum, so $f'(x)$ is negative. - At $x=2$, $f(x)$ has a local minimum, so $f'(x) = 0$ again. - For $x > 2$, $f(x)$ increases sharply, so $f'(x)$ is positive and large again. 4. **Constructing $f'(x)$ graph:** - Plot zeros of $f'(x)$ at $x=-1$ and $x=2$. - Between these zeros, $f'(x)$ is negative. - Outside these zeros, $f'(x)$ is positive. - The magnitude of $f'(x)$ corresponds to the steepness of $f(x)$. 5. **Summary:** The graph of $f'(x)$ crosses the x-axis at points where $f(x)$ has local maxima or minima. **Final answer:** The graph of $f'(x)$ is a curve that crosses the x-axis at $x=-1$ and $x=2$, is positive and large for $x < -1$ and $x > 2$, and negative between $-1$ and $2$.