Derivative of a Real Function on an Interval

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $I$ be an interval containing two distinct points, let $f:I\to\mathbb{R}$, and let $x\in I$.

(i) A real number $L$ is a derivative of $f$ at $x$ if for every real $\varepsilon>0$ there is a real $\delta>0$ such that every $t\in I$ with $0<|t-x|<\delta$ satisfies

$$\Bigl|\frac{f(t)-f(x)}{t-x}-L\Bigr|<\varepsilon ,$$

the quotient being defined because $t-x\neq0$, by Rules of Arithmetic and Order in an Ordered Field (x).

(ii) $f$ is differentiable at $x$ if it has a derivative at $x$. This derivative is then unique, by Uniqueness of the Derivative of a Real Function on an Interval (i), and is written $f'(x)$.

(iii) For a subset $S\subseteq I$, $f$ is differentiable on $S$ if it is differentiable at every point of $S$, and $f$ is differentiable if it is differentiable on $I$.

(iv) If $f$ is differentiable, its derivative is the map $f':I\to\mathbb{R}$, $u\mapsto f'(u)$ (Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (v)).