Real function/Derivative function/Definition

Derivative

Let ${}I\subseteq \mathbb {R}$ denote an interval, and let

$f\colon I\longrightarrow \mathbb {R}$

be a function. We say that ${}f$ is differentiable, if for every point ${}a\in I$, the derivative ${}f'(a)$ of ${}f$ in ${}a$ exists. In this case, the mapping

$f'\colon I\longrightarrow \mathbb {R} ,x\longmapsto f'(x),$

is called the derivative of ${}f$.