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?

Short answer: You are incorrect because when calculating a limit of a function, you only need to look at its domain. So, if a function is defined on $[a,b]$, then its limit at $a$ is equal to its right limit at $a$.
