I am a beginner in derived algebraic geometry and I am trying to develop some visual and geometrical intuition about derived schemes (and stacks), or more precisely about the new geometrical phenomena that they introduce.

Following [Vezzosi's note about the cotangent complex][1] he says that the cotangent complex $\mathbb{L}_{X}$ of some scheme $X$

> does encode (via its Ext-groups)
precisely all deformations of $X$, as long as one interprets these deformations as deformations in *higher derived directions*.

The example he cites after that (the one in "Definition 1.1") is not very clarifying for me: I do not know  the "geometrical shape" or "how to visualize" the derived spectrum of the "*dg-ring of $i$-th order dual numbers over $k$*".

In constrast, an underived scheme, I more or less know how to visualize the information that nilpotents carry (i.e. information about the $n$-derivatives in the multiple points, like for example this illustration (from "*The Geometry of Schemes*" book) for the double point in $(0,0)$)

[![enter image description here][3]][3]

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**QUESTION:** My question would be if someone can give  some geometrical example of these "derived directions" in derived schemes in a similar way than the cited example of a double point in an underived scheme


  [1]: https://arxiv.org/pdf/1008.0601.pdf
  [3]: https://i.sstatic.net/8puQd.png