I'm quite new to this level of calculus, and I'm trying to understand this proof for the equality of mixed partial derivatives:
https://services.math.duke.edu/~wka/math103/newmixed.pdf
I'm fine up to the line ..... =$\frac{\partial A}{\partial y}$(x, $\eta_A$)(y-b) about half way down, but I can't understand how they arrive at the next line (and the next after that, for that matter). When I differentiate $\frac{\partial }{\partial y}($A(x, $\eta_A$)) on its own, I arrive at the required $\frac{\partial }{\partial y}(f$(x, $\eta_A$))-$\frac{\partial }{\partial y}(f$(a, $\eta_A$)) , and so when I differentiate that first line in full, with the inclusion of (y-b) as a product, I get a totally different answer but not sure why?
I'm sure I'm going wildly wrong somewhere, even in as basic a step maybe as misunderstanding the notation? Any insight you could provide would be greatly appreciated, many thanks.
You're totally correct. They have omitted, in error, the multiplicative term $y-b$ [and, similarly, $x-a$ down about six lines], but then it reappears, as it should, in the subsequent line. (This document is in Bill Allard's directory, but he retired quite a few years ago, so I suspect not too many people are looking at it.)