Left Hand Derivative Definition
The left-hand and right-hand derivatives of $f$ at $a$ are defined by $$ f'_{-}(a)=\lim_{h\to 0^-}\frac{f(a+h)-f(a)}{h} $$ and $$ f'_{+}(a)=\lim_{h\to 0^+}\frac{f(a+h)-f(a)}{h} $$ if these limits exist. Then $f'(a)$ exists if and only if these one-sided derivatives exist and are equal.