Sketch derivative branches around two vertical asymptotes

The printed Teacher answer is $x<-2$ or $x>2$. The inputs $x=\pm2$ are vertical asymptotes, not points of the curve; they are excluded. Between them the given curve rises, passing through the origin with a horizontal stationary inflection.

The source Teacher derivative sketch retains vertical asymptotes $x=-2$ and $x=2$. It is negative outside these inputs, positive on $(-2,0)$ and $(0,2)$, and zero at 0. The central branch touches the x-axis at the origin as a local minimum of the derivative, rather than crossing it.

For the drawn derivative, approach $-2$ from the left gives $-\infty$ and from the right $+\infty$; approach 2 from the left gives $+\infty$ and from the right $-\infty$. The outer derivative branches approach $y=0$ from below in both distant tails. These are features of the Teacher sketch, justified by the source curve’s slopes, not by applying an asymptote rule without looking at the graph.

Include every branch on the domain $\mathbb R\setminus\{-2,2\}$, with the two asymptote inputs excluded. The whole relation is not continued through an asymptote. No exact function formula, derivative branch height or additional intercept is supplied.