Lesson 9 of 12
Interactive lesson · derivatives
Limit Definition of the Derivative: Secant to Tangent
Two points determine a secant line and an average slope. Keep one point fixed at x=1 and slide the other point closer. The interval shrinks, and the secant line settles toward the tangent line.
Learning goalConnect average slope over an interval to instantaneous slope at a point.
f(1+h)−f(1)h → 2 as h→0
Move the idea
Try it: move the second point closer
Move the slider
Change one parameter and watch what changes with it.
Instantaneous slope is the limiting value of average slopes over smaller and smaller intervals.
Average slope
The secant slope compares the change in y with the change in x between two distinct points.
Shrink the interval
As h gets smaller, the second point samples the curve more locally around x=1. The secant slope gets closer to 2.
Do not set h=0
The derivative does not divide by zero. It asks what the quotient approaches for nonzero h as h gets arbitrarily small.
Step 3 · One-question check
Did it click?
For f(x)=x² at x=1, what happens to the secant slope as the second point approaches x=1?
Next: read the tangent slope directly as the derivative