Implicit functions - existence, uniqueness, and how to differentiate I have a real analysis textbook which has several chapters dedicated to implicit functions.

*

*implicit functions (of one variable) defined by one equation.


*implicit functions (of several variables) defined by one equation.


*implicit functions (of several variables) defined by several equations (i.e. by a system of equations)
Each of the above chapters teaches us how to differentiate such implicit functions.


*theorem for existence and uniqueness of implicit functions (this one is marked with * so it's harder to read)

This final chapter seems to be very complicated, the proof seems to span 5-6 pages and the conditions in the theorem and the notations are not easy at all to follow.
I have two questions:
a) Is it worth studying the last chapter, if you're just a hobbyist mathematician? After all, what would the proof give me?! I trust the theorem (under its conditions). It is the first proof I intend to skip in this book and that's after going through 400 pages.
b) When calculating the derivatives of implicit functions we treat the implicit functions as known. Why is that so? After all they are implicit so they are not really known.
Just asking for advice here.
 A: Can't address the first, but as for

When calculating the derivatives of implicit functions we treat the implicit functions as known. Why is that so? After all they are implicit so they are not really known.

It's not that an implicit function is "not really known" but that an implicit function generally has no explicit closed-form expression in the variables of the problem. After all, up to technical fine print about the domain and the value at one point, an implicit function is uniquely determined and (under fairly standard hypotheses on the implicit equation) is continuously differentiable.
To make an analogy, in an algebra problem we can call an unknown number $x$. We don't necessarily know $x$ by inspection, but we know it satisfies arithmetical properties. By solving for $x$, we mean finding an equivalent condition, or a necessary condition, that can be solved by inspection.
Similarly, if we call an implicit function $\phi$, we don't miraculously obtain  a formula for $\phi(x)$, but we do know $\phi(x)$ satisfies an equation of the type
$$
F(x, \phi(x)) = 0
$$
for some explicitly-known continuously-differentiable function $F$. Viewing the left-hand side as the value of a function of $x$, we can differentiate using properties of the derivative, obtaining an explicit equation of the form
$$
G(x, \phi(x), D\phi(x)) = 0
$$
that is linear in $D\phi$ and can therefore be solved explicitly for $D\phi$ in terms of $\phi(x)$ and known functions of $x$ by grouping all the "$D\phi$ terms" on one side and all the "non-$D\phi$ terms" on the other.
