World 1 · Functions & Limits · 8 minInstantaneous rate
When the secant becomes the tangent
Teaching this? Open projector mode
What is the speed at a single instant, not over an interval? Take the secant slope and let the second point slide toward the first. As the run shrinks to zero, the secant pivots into the tangent line — and its slope is the instantaneous rate of change.
This limit — the secant slope as the gap → 0 — is exactly the derivative. World 1 has been building the runway; this is the takeoff. You have quietly assembled every idea the derivative needs: functions, slope, and limits.
Check yourself
The tangent slope is the limit of the secant slope as…
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