Set the Point Free and the Slope Becomes a Function

You'll learnto build the derivative as a limit of difference quotients, and to say which notation means a function and which means one number.

Pick a point on a curve. Pick a second point a distance h to its right. The line through the two of them has a slope you can actually compute — rise over run, from coordinates you already have. Now shrink h. The chord swings until it rests against the curve at a single point, and the number it settles on is f′(3). Then do the move that separates 2.2 from 2.1: stop naming the point. The same limit runs at every x at once, and what comes out is not a number any more. It is a function.

8 STEPS · 6 QUICK CHECKS · h → 0, NEVER x → 0 · f′(x) IS A FUNCTION, f′(3) IS ONE NUMBER · v1

f(x) = x² − 4x ⇄ f′(x) = 2x − 4

Before you start

What you're looking at

The parabola f(x) = x² − 4x, with a fixed point at x = 3 and a second point h further along. The line through them is the chord whose slope we can measure.

The question

What number does that chord's slope approach as h shrinks — and what happens if we never name the point at all?

Watch for

The chord rotating into the tangent, the readout converging on 2, and the second panel where the whole derivative function gets drawn while the point sweeps.

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