Applied Mathematics · Ch 8 — Calculus
Instantaneous Rate of Change and Derivative
8.8
Instantaneous Rate of Change and Derivative
We have seen how the average rate of change tells us the slope over an interval, but what if we want the speed of a car at a precise instant, or the slope of a curve at a single point? That is the idea of the instantaneous rate of change — the limit of the average rate as the interval shrinks to zero. This limit is what we call the derivative, and it is the fundamental tool of calculus for d …
Figure 8.8The derivative as the limit of secant slopes: as Q approaches P along the curve, the secant PQ turns into the tangent at P
Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your NCERT textbook's own diagram.
As Q slides toward P, the secant PQ approaches the tangent at P — the derivative is that …