So I've just started to get into some calculus and I recently came across the topic of implicit differentiation. I am extremely confused on what implicit functions are and there is very little information on what exactly implicitly defined functions really are but instead lots of defintions on how to implicitly differentiate.
I became even more confused from this definition proofwiki provides which defines an implicit function as follows.
Consider a (real) function of two independent variables $z=f(x,y).$
Let a relation between $x$ and $y$ be expressed in the form $f(x,y)=0$ defined on some interval $I$.
If there exists a function:
$y=g(x)$ defined on $I$ such that:
$∀x∈I:f(x,g(x))=0$ then the relation $f(x,y)=0$ defines $y$ as an implicit function of $x$.
I can't seem to wrap this definition i am familiar with how a function is a special kind of binary relation but when we state that let a relation between $ x, y$ be expressed in the form $f(x,y)=0$ what exactly do we mean by this for example how is $x^2 +y^2=1$ a relation in that form. a simple example would be greatly appreciated.
I hope this question is not too vague to be answered or too silly to be even considered thanks in advance.