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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. Its behaviour as the two points come together is the whole question: it may settle on a number, on two different numbers, or on none.

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

Secants closing on the tangent to x². Secant lines through x = 1 and a second point 1.2, 0.8, 0.5, 0.28, 0.12 away, with the slope of each. They approach 2, the derivative there.

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 6 cosines, and a curve with no tangent anywhere. Partial sums of a sum of cosines whose amplitudes shrink geometrically and whose frequencies grow faster. Each term adds finer detail; the curve converges and its slopes do not.

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 A rectangle grown on two sides. A rectangle x by √x, with both sides grown by the change a step of h makes. The new area is the old one, two strips, and a small corner rectangle that has both increments in it.

A rectangle grown on two sides

A product of two changing quantities is the area of a rectangle whose sides both move. The extra area is two strips and a corner, and the whole of the product rule is the observation that the corner is negligible and the strips are not.

analysis · The derivative

Named alongside it

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

ContinuityDerivativeDifferentiabilityLimitApproximationAreaConvergenceCounterexampleDissectionFourier seriesFractal dimensionIntegral

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