As the gap between two points shrinks, a secant becomes a tangent
The slope of the line (secant) joining two points on a curve is the average rate of change. Squeeze the gap h between the two points toward 0 with a slider, and watch with your own eyes what the instantaneous rate of change (the derivative) turns out to be, as the secant turns into a tangent.
If a car drives 100km in 2 hours, its average speed is 50km/h โ but "the speed right at this instant" is a different question. The slope of the line (a secant) joining two points on a curve is the average rate of change, but squeeze the gap h between the two points infinitely close to 0, and the secant turns into a tangent that just barely touches the curve at that one point.
The limit value at that point is the instantaneous rate of change (the derivative). For f(x)=xยฒ, the instantaneous rate of change is 2 at x=1, and 4 at x=2 โ always exactly double the original x. Turn this rule into a function, and you get the derivative f'(x)=2x.
Use the slider to move h toward 0 and watch directly as the secant turns into a tangent, then compare the graphs of the original function and its derivative side by side.
Average rate of change โ The slope of the secant joining two points A=(1, f(1)) and B=(1+h, f(1+h)) on f(x)=xยฒ is (f(1+h)โf(1))รทh. Try shrinking h very close to 0. Watch the secant (violet) gradually settle onto the tangent (blue dashed line).
Secant AB (average rate of change)Tangent (instantaneous rate of change)
h (B is at 1+h)1.00
Slope = (f(1+h) โ f(1)) รท h
The derivative f'(x) is a new function made by gathering up the tangent slope at every point on the curve. Move x with the slider, and you can see that how steeply the tangent (in the top graph, f) tilts always matches exactly the height of the bottom graph (f') at that same x.