How is applications of differentiation related to calculus of variation?

an area of development in Calculus of variation that is still ongoing.
application of use of the context of double soap bubble.
key equaion or theorem underlying study of calculus of variation.
an example of another field of study that uses calculus of variation that has an impact on society.

In Calculus of Varations deals with minimization problems the most elementary of which is the Brachistochrone. This is the problem of the shape of a wire which provides the fastest path between two points for a marble sliding down it. I don't know what applications of differentiation is.

I don't understand your question, you may want to try rephrasing it.

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