Definition:Differentiable Mapping/Real Function/Point

Definition
Let $f$ be a real function defined on an open interval $\left({a \,.\,.\, b}\right)$.

Let $\xi$ be a point in $\left({a \,.\,.\, b}\right)$.

Definition 2
This limit, if it exists, is called the derivative of $f$ at $\xi$.

Also see

 * Equivalence of Definitions of Differentiable Real Function at Point