Intuition behind the difference between derived sets and closed sets? I missed the lecture from my Analysis class where my professor talked about derived sets. Furthermore, nothing about derived sets is in my textbook. Upon looking in many topology textbooks, few even have the term "derived set" in their index and many books only say "$A'$ is the set of limit points of $A$". But I am really confused on the difference between $A'$ and $\bar{A}$. For example,
Let $A=\{(x,y)∈ \mathbb{R}^{2} ∣x^2+y^2<1\}$, certainly $(1,0) \in A'$, but shouldn't $(0,0)∈A'$ too? In this way it would seem that $A \subseteq A'$.
The definition of a limit point is a point $x$ such that every neighborhood of $x$ contains a point of $A$ other than $x$ itself. Then wouldn't $(0,0)$ fit this criterion? If I am wrong, why? And if I am not, can someone please give me some more intuitive examples that clearly illustrate, the subtle difference between $A'$, and $\bar{A}$?

In the example you give, the derived set is indeed the same as the closure. However, that need not always be the case. 
Let's start with a silly example. Consider the set $A=\{17\}$. This is a closed set, so $\overline{A}=A$. But $A$ has no limit points, so $A'=\emptyset$.
Here is a slightly more complicated example. Let $A$ be the set of all numbers of the shape $\frac{1}{n}$, where $n$ ranges over the positive integers. Then the closure of $A$ is just $A$ together with the number $0$. But the derived set of $A$ is just $\{0\}$.
