1. The problem asks us to sketch the derivative of a graph with a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$. 2. The original graph behavior is: - For $x<0$, the curve is negative and decreases steeply towards negative infinity near $x=0$. - For $x>0$, the curve is positive and decreases steeply towards positive infinity near $x=0$. 3. To find the derivative's shape, recall that the derivative represents the slope of the tangent line at each point on the original graph. 4. Near $x=0$, the original graph shoots steeply down to negative infinity on the left and up to positive infinity on the right, so the slope magnitude is very large. 5. For $x<0$, the graph is decreasing steeply, so the derivative is large negative near zero from the left. 6. For $x>0$, the graph is decreasing steeply but positive, so the slope is large negative near zero from the right as well. 7. Away from zero, the graph approaches zero horizontally, so the slope approaches zero. 8. Therefore, the derivative graph has a vertical asymptote at $x=0$ with the derivative going to negative infinity on both sides near zero and approaching zero far from zero. 9. Among the options, option D matches this description: vertical asymptote at $x=0$, sharp dip near zero going to negative infinity on both sides, and approaching zero at the extremes. Final answer: The derivative graph corresponds to option D.