I want to highlight a potential ambiguity, which is that “Newton’s approximation” is sometimes used to mean Newton’s method for finding roots, but the “Newton’s approximation” I had in mind is the one given in Tao’s Analysis I, Proposition 10.1.7, which is a way of restating the definition of the derivative. (Here is the statement in Tao’s notes in case you don’t have access to the book.)
I want to highlight a potential ambiguity, which is that “Newton’s approximation” is sometimes used to mean Newton’s method for finding roots, but the “Newton’s approximation” I had in mind is the one given in Tao’s Analysis I, Proposition 10.1.7, which is a way of restating the definition of the derivative. (Here is the statement in Tao’s notes in case you don’t have access to the book.)
Ah that makes sense, thanks. I was in fact thinking of Newton’s method (which is why I didn’t see the connection).