From secant to tangent

The derivative is what a secant becomes. Walk the five steps, then drag h until the amber line lies on the oxblood tangent.

1. 01

   Fix a point on the curve

   The derivative at a is a slope at one x-value. Pick a, read f(a), and leave that point still while everything else moves.

2. 02

   Draw a secant with a second point

   The second point sits at a + h. The amber line through both points is a secant. Its slope is the difference quotient, [f(a+h) - f(a)] / h.

3. 03

   Shrink h, do not set it to zero yet

   The definition is a limit. Drag |h| down. The secant pivots. The readout should be walking toward the number you already know is f'(a).

4. 04

   Name the tangent as that limit

   The oxblood ghost is the tangent, slope f'(a). When the secant lies on it, you have watched the limit. The derivative is that slope, not a separate object.

5. 05

   Try a corner in your head

   If the two one-sided secants refused to agree, the limit would fail and there would be no tangent. That is why |x| is not differentiable at 0.

Visualizer page: secant to tangent. Tracer: tangent line tracer.