The Knowledge Web

The Paradox

To calculate a slope (velocity), you need two points: a start and an end.

m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}

But for Instantaneous Velocity, we are looking at a single moment in time. The start point is the end point. If we try to plug this into the slope formula, we get disaster:

m = \frac{f(x) - f(x)}{x - x} = \frac{0}{0}

The Solution: The Secant Line

We can't use one point. So we use two points... and cheat. We place a second "fake" point at distance $h$ away from our target. Then we shrink $h$ until it disappears.

The Secant Slider

SECANT LINE

Run (Δx)2.0000

Rise (Δy)2.0000

Slope1.0000

True Derivative0.5000

Shrink Distance (h)2.00

0 (Tangent)4 (Secant)

Drag the slider left to pull the blue point closer to the red anchor.

Drag

h

to zero. Watch the blue Secant line become the red Tangent line.

The Formal Definition

This logic leads us to the most famous equation in Calculus: The Definition of the Derivative.

f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

Let's decode this:

$f(x+h) - f(x)$ The "Rise" (change in y)
$h$ The "Run" (change in x)
$\lim_{h \to 0}$ The "Shrink Ray" that pushes the points together.