Answer
$f(x_{0})=\lim\limits_{x \to x_{0}}\frac{f(x)-f(x_{0})}{x-x_{0}}$
$f'_{-}(x_{0})=\lim\limits_{x \to x_{0^{-}}}\frac{f(x)-f(x_{0})}{x-x_{0}}$
$f'_{+}(x_{0})=\lim\limits_{x \to x_{0^{+}}}\frac{f(x)-f(x_{0})}{x-x_{0}}$
Work Step by Step
The derivative of a function is represented as:
$f(x_{0})=\lim\limits_{x \to x_{0}}\frac{f(x)-f(x_{0})}{x-x_{0}}$
The above equation can be represented as left and right handed derivatives:
$f'_{-}(x_{0})=\lim\limits_{x \to x_{0^{-}}}\frac{f(x)-f(x_{0})}{x-x_{0}}$
$f'_{+}(x_{0})=\lim\limits_{x \to x_{0^{+}}}\frac{f(x)-f(x_{0})}{x-x_{0}}$
When both the left and right handed derivatives exist and give the same value, then the derivative of the function exists and is equal to that value.
