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

Graphical or geometric meaning

On a position-time graph, velocity is tangent slope. A negative slope means position is decreasing. On a velocity-time graph, acceleration is tangent slope, and signed area under velocity over a time interval gives displacement.

Common mistakes and why they fail

Verification and reasonableness checks

• Estimate tangent slope from nearby secant slopes.
• Use dimensional analysis on every derivative.
• Check whether the sign agrees with the graph's local direction.

Average rates do not determine instantaneous rates

An average rate on $[a,b]$ is a secant slope; an instantaneous rate is the limit of secant slopes near one input. They can coincide, but need not. Units are a strong diagnostic: position in metres and time in seconds give velocity in metres per second, whereas acceleration uses metres per second squared. A negative velocity means motion in the negative coordinate direction, not negative speed; speed is $|v|$. Interpret a derivative by stating the time, units, and direction. With discrete data, a centered secant slope often gives a better local estimate than a one-sided slope using the same spacing, because it samples behavior on both sides.

Practice

1. For $s=t^2+1$, find velocity at $t=3$.
2. If $C(q)=q^2+10$, interpret $C'(5)=10$.
3. If $v(4)=-2$, what is speed?

Connections and next steps

• Derivative
• Derivative from First Principles
• Related Rates