This question is sort of similar to my second question here: Proof of the Implicit Function Theorem in Baby Rudin. But since that question hasn't been answered yet, I decided to post this one.
How did Rudin compute the highlighted derivative? Are there any differentiation rules for maps $E\subset \mathbb R^n\rightarrow \mathbb R^n$ which he does not mention explicitly? Or is this some triviality?
On the intuitive level it is clear that this holds, if, by analogy with a real-valued function of a real variable, one treats $x$ as the variable and $y$ as a constant, but we have a rigorous definition of a derivative of a map $\varphi: E\subset \mathbb R^n\rightarrow \mathbb R^n$ at $x$ (namely a linear operator $L: \mathbb R^n\rightarrow \mathbb R^n$ such that $\lim_{v\rightarrow 0}\frac{|\varphi(x+v)-\varphi(x)-Lv|}{|v|}=0$), and I don't see why this holds; and even if this holds, I how can one find it (other than drawing analogy with 1 variable)?
