derivative on endpoints what's the derivative of $f(x)= x^{2}$ ($x\geq 0$) when x=0?
from my understand, it doesn't exist because even $\lim_{h \to 0^{+}}\frac{f(x+h)-f(x)}{h}$ is 0, but  $\lim_{h \to 0^{-}}\frac{f(x+h)-f(x)}{h}$ doesn't exist, am I correct?

If you look for the rigorous definition of limit, then you may understand that $f'(0)$ exists. In high-school or calculus course, we usually learn limit naively because rigorous approach is too hard. But sometimes it occurs cognitive obstacles, and thinking that limit exists only if left and right limit must exist like you is also cognitive obstacles. Let's introduce rigorous definition of limit of function- Let $A\subset \mathbb{R}$ be a domain of $f$.

If for every $\epsilon > 0$, there exists $\delta > 0$ such that if for all $x\in A$, $|x-a|<\delta$ then $|f(x)-L|<\epsilon$, then we say $f$ converges to $L$ as $x\to a$.

Then how about left or right sided limit?

If for every $\epsilon > 0$, there exists $\delta > 0$ such that if for all $x\in A$, $a < x <a+\delta\;(a-\delta < x < a)$ then $|f(x)-L|<\epsilon$, then we say that $L$ is a right(left) sided limit of $f$.

How about $f(x)=x^2\;(x\ge 0)$? Here $A=[0,\infty)$, then if $x\in (-\delta,0)\cup(0,\delta)$ and $x\in A$, then $A\cap (-\delta,\delta)=(0,\delta)$, thus $x\in (0,\delta)$. Thus right sided limit is enough.
