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 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 $(0,0)$)

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

EDIT (Sorry for being vague and doing mistakes in the following but I am a beginner trying to get some intuition)

I am a bit unsure about how the fact of derived geometry being "formal" hampers thinking about it in some spatial/visual sense.

For example, in the case of my initial question (Definition 1.1. of Vezzosi's note ) I guess you can say that there is nothing more (not extra points nor infinitesimal stuff) in the underived space generated by $k$, but there is some other "space" (which takes into account the extra terms $k[i]$) in wich there is located this infinitisemal disk $\mathbf{D}_{i}=\mathbb{R}\mbox{Spec}k\left [ \varepsilon _{i} \right ] $. My problem comes with the interpretation of these $k[i]$'s or in general any negative term of the cotangent complex as (something that generates) derived directions. I would like to see examples of proper geometrical stuff happening inside the singularities of the underived space and that the shape of those geometries are described by some object in the total space, (the space generated by the whole negative part of the cotangenet complex) . But for that I would need some constraints in forms of morphisms (in the same way that some polynomials define an affine variety) between the terms of the complex to define those shapes in the derived spaces (otherwise I only see loop spaces and some thickenings of difficult interpretation*) , but I cannot find or understand such examples.

*See the note in M.Anel's "The Geometry of Ambiguity" about how

derived infinitesimal thickenings encompasses the limit of the intersection of two lines (say in the plane) when the lines become the same.

(also higher dimensional objects?)

On the other side I think that thinking in the derived structure at the same time that in the stacky structure should be illuminating. For example in a higher stack, say a $2$-vector bundle over a smooth projective variety I can think in an enlarged space in the sense that there is not one bundle over the variety but a larger set of "bundles" related by morphisms and higher morphisms over the same point "at the same time" (in this sense to interpret the higher categorical morphims as degrees of freedom, or "directions" sounds reasonable to me). It is obviously a bad description but it is a first step, however in the derived vector bundle... what the derived structure tell me in geometric terms?... I guess the deformation/obstruction space of it... but I arrive to the same problem: seems to unconstrained to me and there will be different derived extensions for the initial underived stack of vector bundles...

M.Kapranov says in this article that

In some intuitive sense, the positive (right-hand) range corresponds to“geometry in the small”, where we focus on small details near a complicated (singular) point of a geometric object. The negative (left hand) range corresponds similarly to “geometry in the large”, where we focus on large scale, topological properties of spaces

I dont know how much precise is this analogy cause a point in a (higher) stack can also be thought to have a very large set of (higher) symmetries and I guess not all these symmetries/transformations are of large-scale/topological nature but something more abstract (see for example the different definitions of 2-vector spaces). However seems a useful analogy in some sense, but still problematic to me when I try to think in objects having at the same time stacky and derived degrees of freedom. Anyone can explain Kapranov's analogy in more detail?