Derivative as Rate of Change

Average velocity. Suppose a particle moves along a straight line, and

its position (the distance covered, measured from a fixed starting point) at time $t$ is given

by a function $s(t)$. Over a time interval from $t$ to $t+\Delta t$, the particle's position

changes by $\Delta s = s(t+\Delta t)-s(t)$, and its average velocity over that interval is

the change in position divided by the change in time:

This number depends on the whole interval $[t,\,t+\Delta t]$ chosen -- it says nothing about how

fast the particle was moving at any single instant within that interval.

Instantaneous velocity. To find the velocity at the single instant $t$, shrink the

interval by letting $\Delta t\to0$: the average velocity over smaller and smaller intervals

around $t$ settles down to a limiting value, called the instantaneous velocity at time $t$:

This is precisely a limit of the same shape studied throughout this chapter -- a difference

quotient whose limit is taken as the increment shrinks to zero.

Generalising beyond distance. Nothing in the argument above actually used the fact that $s$

measured distance. For any function $y=f(x)$ of a variable $x$ -- position, temperature,

population, area, or anything else expressible as a function -- the same limiting process

defines the instantaneous rate of change of $y$ with respect to $x$ at a point $x$:

This quantity is called the derivative of $f$ at $x$, exactly what Section 8 defines

formally and gives the standard notation $f'(x)$ or $\dfrac{dy}{dx}$. The distance-velocity pair

is simply the single most familiar physical illustration of a derivative -- not a separate idea,

but the same limiting ratio applied to one particular choice of $f$.

Average rate of change, in general. For any function $y=f(x)$, the average rate of change of

$y$ with respect to $x$ over an interval from $x$ to $x+\Delta x$ is

mirroring average velocity exactly; letting $\Delta x\to0$ turns this average into the

instantaneous rate of change (the derivative) at the single point $x$, exactly as letting

$\Delta t\to0$ turned average velocity into instantaneous velocity. …