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Concept

Differentiability

The property of having a tangent at a point, meaning the chord slopes settle on a single value as the two points come together. It is strictly stronger than continuity, and it is what licenses replacing a curve by a straight line near a point.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

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 the patch(0.9, 0.45)its image, and the flat fit1.8-0.90.91.8the matrix of partial derivativesit multiplies area by 4.050the smallest patch drawn, by 4.051patchwidest gap0.60001.80e-10.30004.50e-20.15001.13e-20.07502.81e-30.03757.03e-4(x² − y², 2xy) at (0.9, 0.45), on a patch 0.60 across. The straight grid on the left is carried over by the map on the right, where the dashed grid is theimage under the one linear map that matches it at the pointthe widest gap between the two is 0.1800 here and 7.03e-4 on a patch 16 times smaller — a quarter at each halving, while the patch itself halves

The flat map that fits closest

A derivative is usually met as a number, which works because a line through a point is described by one. In more than one dimension the object that plays the same role is a linear map, and the number was always a one-by-one instance of it.

analysis · the derivative

Named alongside it

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

ContinuityDerivativeDifference quotientLimitApproximationBasisConvergenceCounterexampleDeterminantFourier seriesFractal dimensionLinearity

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