What Are the Different Types of Derivatives in Calculus?

[observer1]

1. Derivatives in first year calculus
2. Gateaux Derivatives
3. Frechet Derivatives
4. Covariant Derivatives
5. Lie Derivatives
6. Exterior Derivatives
7. Material Derivatives

So, I learn about Gateaux and Frechet when studying calculus of variations
I learn about Covariant, Lie and Exterior when studying calculus on manifolds
I learn about Material derivatives when studying fluid mechanics

But I would have never thought there would be so many.

Is this about all there is?

May I ask if someone can categorize these?
When are they used? How are they used? Are there more?
Can you possibly create a table? Are there subsets of each other?
Are some a manifestation of only physics or engineering or math?

Can anyone bring order out of this chaos for me?

 

[fresh_42]

I'm inclined to make an Insight out of it. Looks like an invitation to a small essay.

 

[observer1]

I'm inclined to make an Insight out of it. Looks like an invitation to a small essay.

Please...

 

[fresh_42]

Please...

But this takes some time. Maybe someone is faster here. But I doubt there is a short answer, except perhaps: continuous generalizations.