Upgrade to ADTypes v1

[gdalle]

Supersedes #412

Checklist

• [x] Appropriate tests were added
• [x] Any code changes were done in a way that does not break public API
• [ ] All documentation related to code changes were updated
• [x] The new code follows the contributor guidelines, in particular the SciML Style Guide and COLPRAC.
• [x] Any new documentation only uses public API

Additional context

Add any other context about the problem here.

[codecov[bot]]

Codecov Report

Attention: Patch coverage is 96.87500% with 1 lines in your changes are missing coverage. Please review.

Project coverage is 85.70%. Comparing base (17bb8a8) to head (844d78c).

Files	Patch %	Lines
src/internal/jacobian.jl	75.00%	1 Missing :warning:

Additional details and impacted files

```diff @@ Coverage Diff @@ ## master #426 +/- ## ========================================== - Coverage 86.30% 85.70% -0.60% ========================================== Files 47 47 Lines 2913 2904 -9 ========================================== - Hits 2514 2489 -25 - Misses 399 415 +16 ```

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[gdalle]

I don't understand what the formatting check does? It doesn't seem related to JuliaFormatter

[ChrisRackauckas]

It is JuliaFormatter. It's saying you haven't ran the formatter on this.

[gdalle]

@avik-pal this is starting to look decent in my view

[gdalle]

Thanks for the merge! Can we tag a new release?

https://github.com/SciML/NonlinearSolve.jl/pull/429