One-sided derivative

Definition

Let $f$ be a real function defined on an interval $[a, a + r)$ with $r > 0$. The right derivative of $f$ at $a$ is the one-sided limit $$f'_+(a) = \lim_{h \to 0^+} \frac{f(a + h) - f(a)}{h},$$ when it exists. The left derivative $f'_-(a)$ is defined in the same way, for $f$ on $(a - r, a]$ and $h \to 0^-$.

Remarks

A function on an open interval is differentiable at $a$ if and only if both one-sided derivatives exist and are equal. Their common value is the derivative. One-sided derivatives give a meaning to derivatives at the endpoints of a closed interval. For $|x|$ at $0$, the right derivative is $1$ and the left derivative is $-1$. Mathlib expresses them with HasDerivWithinAt on a half line.

Explanations

L1 human

A one-sided derivative only looks at one side of the point. At a corner of a graph, the slopes from the left and from the right exist but differ.