How to define implicit differentiation for non-functions?
I want to refer to a similar question was asked here
am studying real analysis from the book “Introduction to Real Analysis” by Robert G. Bartle and Donald R. Sherbert. I learned the basic definition of differentiation for functions, which is:
$\lim_{x \to c} \frac{f(x) -f(c) }{x-c} =f'(c)$
However, I do not understand how to apply this definition to non-functions, such as curves or relations, And the implicit differentiation is not even mentioned in the book.
I know that there is a method called implicit differentiation, which uses the chain rule to find the derivative of one variable with respect to another. and answers to this question suggest the use of chain rule which is a bit weird and I do not understand why this method works. The chain rule is proved by using the Carathéodory theorem, which assumes that the function is continuous.