Definition:One-Sided Derivative

Definition

A one-sided derivative is a right-hand derivative or a left-hand derivative.

Right-Hand Derivative

Let $f: \R \to \R$ be a real function.

The right-hand derivative of $f$ is defined as the right-hand limit:

$\ds \map {f'_+} x = \lim_{h \mathop \to 0^+} \frac {\map f {x + h} - \map f x} h$

If the right-hand derivative exists, then $f$ is said to be right-hand differentiable at $x$.

Left-Hand Derivative

Let $f: \R \to \R$ be a real function.

The left-hand derivative of $f$ is defined as the left-hand limit:

$\ds \map {f'_-} x = \lim_{h \mathop \to 0^-} \frac {\map f {x + h} - \map f x} h$

If the left-hand derivative exists, then $f$ is said to be left-hand differentiable at $x$.

Examples

Absolute Value Function at $x = 0$

Let $f$ be the real function defined as:

$\map f x = \size x$

where $\size x$ denotes the absolute value function.

Then:

\(\ds \map {f'_+} 0\)

\(=\)

\(\ds 1\)

where $\map {f'_+} 0$ denotes the right-hand derivative of $f$ at $x = 0$

\(\ds \map {f'_-} 0\)

\(=\)

\(\ds -1\)

where $\map {f'_-} 0$ denotes the left-hand derivative of $f$ at $x = 0$

while the derivative of $f$ at $x = 0$ does not exist.