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Concept

Difference quotient

The change in a function divided by the change in its input, which is the slope of a chord and the thing a derivative is a limit of.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as differentiability — the same set of essays touches all of them, so they are one junction rather than several.

0.511.522.512345xh = 1.2 slope 3.2000h = 0.8 slope 2.8000h = 0.5 slope 2.5000h = 0.28 slope 2.2800h = 0.12 slope 2.1200

The slope of a single point

A slope needs two points. A derivative is the slope at one. The construction that bridges the gap is a sequence of secants, and the whole difficulty of calculus is in what "the limit of that sequence" is allowed to mean.

analysis · the derivative -0.4-0.20.20.4-0.50.511.52xthe sum1 term2 terms3 terms6 termseach term has 0.5 times the height of the one before and 3 times the frequency, so ab = 1.5 — above one, and thatis the condition6 terms are drawn, the last at 1,215 samples for its 122 oscillations; the sum converges because the heights area geometric series, and the slopes do not because they are multiplied by 1.5 at each step

A curve with a corner at every point

Continuity means a curve can be drawn without lifting the pen. Differentiability means it has a tangent. The first was assumed to nearly imply the second until 1872, when Weierstrass exhibited a curve that is continuous everywhere and has a tangent nowhere — and it is a sum of cosines.

analysis · the derivative

Named alongside it

The objects these essays reach for when they reach for this one.

ContinuityDifferentiabilityConvergenceCounterexampleDerivativeFourier seriesFractal dimensionLimitSecantSelf similarityUniform convergenceWeierstrass function

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