One-Sided Derivative/Examples/Absolute Value Function at Zero

Examples of One-Sided Derivatives

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.

Proof

Demonstrated in Derivative of Absolute Value Function.

$\blacksquare$

Sources

• 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 2$. Functions of One Variable: $2.2$ Derivatives
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): left and right derivatives