The Left Handed derivative of $f$ at $a$ is a number, denoted $f'\_(a)$, which is defined as  $$f'\_(a) = \lim_{x \rightarrow a^-} \frac {f(x)-f(a)}{x-a}$$ or  $$f'\_(a) = \lim_{h \rightarrow 0^-} \frac {f(a+h)-f(a)}{h}$$.
The ratios used in the limit expressions can be interpreted the same as the derivative except that the limit is considered only from values less than $a$.

The Right Handed derivative of $f$ at $a$ is a number, denoted $f'_+(a)$, which is defined as  $$f'_+(a) = \lim_{x \rightarrow a^+} \frac {f(x)-f(a)}{x-a}$$ or  $$f'_+(a) = \lim_{h \rightarrow 0^+} \frac {f(a+h)-f(a)}{h}$$.
The ratios used in the limit expressions can be interpreted the same as the derivative except that the limit is considered only from values greater than $a$.


On the mapping diagram a one sided derivative can be visualized using focus points for the average rates that converge to a single focus point from only one side of the key number of interest for the best linear approximating function for that side.
A one sided derivative can also be visualized as a vector on the target of the mapping diagram.
In the figure the vector shown is for $f'_+(1)$.
For $x \ne 1$, in the example in the figure, the one sided derivatives are equal, so $f'_+(x)=f'\_(x)=f'(x)$.