One-Sided Derivatives

Definition

• Right-hand derivative at $a$: $\lim_{h \to 0^+} \frac{f(a + h) - f(a)}{h}$
  • How to read: “The limit as h approaches zero from the right of the quantity f evaluated at a plus h, minus f evaluated at a, all divided by h.”
  • Meaning: The instantaneous rate of change of $f$ at $a$ when approached only from the right (values $x > a$).
• Left-hand derivative at $b$: $\lim_{h \to 0^-} \frac{f(b + h) - f(b)}{h}$
  • How to read: “The limit as h approaches zero from the left of the quantity f evaluated at b plus h, minus f evaluated at b, all divided by h.”
  • Meaning: The instantaneous rate of change of $f$ at $b$ when approached only from the left (values $x < b$).

Why It Matters

One-sided derivatives are the mathematical tools for dealing with “The Sudden.” In a world where systems are usually smooth, one-sided derivatives handle the “kinks”—the exact moment a ball hits a wall, a market crashes, or a circuit breaker trips. Without them, we couldn’t define the physics of collisions or the behavior of piecewise systems. They allow us to calculate the “slope” of a change even when that change is interrupted, ensuring that our models don’t break down just when things get interesting.

Core Concepts

In the context of calculus, a function is differentiable on a closed interval if it has derivatives at the interior points and appropriate one-sided derivatives at the endpoints.

• Differentiability at a Point A function has a (two-sided) derivative at an interior point $x = c$ if and only if both the left-hand and right-hand derivatives exist at $c$ and are equal: $f'(c) \text{ exists} \iff f'_-(c) = f'_+(c)$
  • How to read: “The derivative of f evaluated at c exists if and only if the left-hand derivative at c is equal to the right-hand derivative at c.”
  • Meaning / when to use: A sharp corner or jump in slope (like $|x|$ at 0) means the two-sided derivative does not exist even if each one-sided slope is finite.

For example, $f(x) = |x|$ has a right-hand derivative of $1$ and a left-hand derivative of $-1$ at $x = 0$. Since they are not equal, $f(x) = |x|$ is not differentiable at $x = 0$.

• How to read: “The right-hand derivative of f at zero is equal to one; and the left-hand derivative of f at zero is equal to negative one.”
• Meaning: The canonical example of a continuous function that is not differentiable at a kink—one-sided slopes disagree at $x = 0$.

Connected Concepts

• One-Sided Limits
• One-to-One Functions
• Inverse Functions
• Directional Derivatives