The Derivative and the Paradox of Instantaneous Change

What does it mean to describe change at a single instant? A derivative gives a precise mathematical answer by taking a limit of rates measured over nonzero intervals.

Imagine a car starting at A, speeding up and then stopping at B, 100 meters away after ten seconds. Plot distance s(t) against time: the curve is shallow when the car is slow and steeper when it travels farther per second.

A corresponding velocity graph rises and falls. The question is how the velocity at a time relates to the distance curve, given that an average rate normally compares two different times.

As a simplified measurement example, take the distance traveled from t to t + Δt and divide by Δt. This is an average velocity over a short interval, not yet the exact derivative. The video notes that actual speedometers commonly use other physical mechanisms.

On the graph, Δt is the horizontal step and Δs the vertical change. Their ratio is the slope of a secant line through two nearby points. Repeating this calculation for many times gives an approximation of the velocity curve.

The derivative is the limit of [s(t + Δt) − s(t)] / Δt as nonzero Δt approaches zero, provided that limit exists. It is not obtained by setting Δt to zero in the original quotient.

Geometrically, as the two points come together, the secant slopes approach the tangent slope when the function is differentiable. This offers a visual interpretation of the limit.

Sanderson suggests thinking of the derivative as the best local linear approximation to change. “Instantaneous rate of change” is standard shorthand for this well-defined limit, not a claim that a finite change occurs within a duration of zero.

The video uses ds and dt for small changes while introducing the idea, then writes ds/dt for the limiting derivative. In formal work, Δs/Δt can distinguish a finite difference quotient from derivative notation.

For s(t) = t³, the difference quotient at t = 2 is [(2 + Δt)³ − 2³] / Δt. Expanding the cube and canceling the nonzero Δt gives 12 + 6Δt + (Δt)².

As Δt approaches zero, the remaining terms containing Δt approach zero, leaving 12. More generally, the derivative of t³ is 3t², so the velocity at t = 2 is 12 in the example’s units.

The algebra is a derivation of a reusable rule: once the power rule is known, one need not expand the difference quotient from scratch for every calculation.

At t = 0, the derivative of t³ is zero, even though the car moves over any positive interval immediately afterward. A zero derivative describes the local first-order rate at that instant; it does not imply the function stays constant in a neighborhood.

For example, from 0 to 0.1 seconds, the distance changes by 0.001 meters and the average speed is 0.01 meters per second. As the interval shrinks, that average tends to zero, consistent with the derivative.