Derivative as a Rate of Change

The central idea

If $s(t)$ is position, the average velocity on $[t,t+h]$ is $[s(t+h)-s(t)]/h$ and the instantaneous velocity is $v(t)=s'(t)=\lim_{h\to0}[s(t+h)-s(t)]/h$. More generally, $y'=dy/dx$ measures instantaneous output change per unit input. Acceleration is $a(t)=v'(t)=s''(t)$.

Definitions, hypotheses, and notation

An instantaneous rate is local but can predict a nearby finite change: if $Q'(a)=r$, then $Q(a+h)-Q(a)\approx rh$ for small $h$. This approximation is first order; curvature controls how quickly it loses accuracy. In applications, the derivative can be marginal cost, density per unit length, population growth per year, or sensitivity to a parameter. In every case its units are output units divided by input units.

For motion, direction and speeding up are separate questions. A particle speeds up when velocity and acceleration have the same sign and slows down when their signs differ. A zero velocity may mark a turnaround, but only if velocity changes sign. A zero acceleration marks no instantaneous change in velocity; it says neither that velocity nor position is zero.

Conceptual meaning

A derivative carries units. If distance is metres and time seconds, velocity is metres per second and acceleration is metres per second squared. Its sign describes direction of change, while its magnitude describes sensitivity or speed.

A dependable method and decision rule

1. Name the changing quantities and their units.
2. Write the relevant function before differentiating.
3. Differentiate symbolically, then evaluate at the requested input.
4. Distinguish the derivative's sign from the original quantity's sign.
5. Translate the numerical result into a complete sentence with units.

Fully worked example

Common mistakes and why they fail