Add docstrings and update documentation

[gdalle]

Fix #144, fix #181

• Remove function signatures from README.md and add them as docstrings
• Structure all docstrings on the docs/src/api.md page
• Remove the additional comments from README.md and put them in the proper section of docs/src/api.md
• Remove the tutorials from README.md and turn them into docs/src/tutorials.md (no doctests yet)
• Copy-paste both README.md and docs/src/reproducibility.md into docs/src/index.md

I didn't manage to update Documenter to v1.3 (it complained about the reproducibility part), so I'm leaving it for a later PR

[gdalle]

It looks like the downstream test fail on 1.6 has nothing to do with me? I didn't change the source code only the docstrings

[ChrisRackauckas]

Thanks!

It looks like the downstream test fail on 1.6 has nothing to do with me? I didn't change the source code only the docstrings

We'll make this repo just follow the standard deprecation to LTS here.

[gdalle]

awesome! can a new release be tagged so that the cool doc becomes available?

[ChrisRackauckas]

Already initiated.

[gdalle]

Is it expected that the new documentation doesn't show up on https://docs.sciml.ai/FiniteDiff/stable/ nor https://juliadiff.org/FiniteDiff.jl/stable/?
And which one of those is the right URL?

[ChrisRackauckas]

It needs to wait for the next global build