Physical Meaning of the Derivative
Big Idea: The derivative isn't just an abstract mathematical concept — it measures how fast something is changing at any given instant. It connects algebra and calculus to the real, physical world.
1. From Average Rate to Instantaneous Rate
Imagine you're driving a car. Your speedometer doesn't show your average speed over the whole trip — it shows your speed right now, at this very instant. That's exactly what a derivative captures.
Average Rate of Change
If a function $s(t)$ gives position at time $t$, then the average velocity over the interval $[t, t+\Delta t]$ is:
$$v_{\text{avg}} = \frac{s(t + \Delta t) - s(t)}{\Delta t} = \frac{\Delta s}{\Delta t}$$
This is the slope of the secant line between two points on the graph of $s(t)$.