The derivative — the series
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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.
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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.
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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.
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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.
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The slope of the mirror image
Undoing a function is reflecting its graph in the diagonal, and a reflection turns a slope into its reciprocal. That single observation supplies the derivative of every inverse — the logarithm, the roots, the inverse trigonometric functions — without differentiating any of them.
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A slope for a curve that is no function
The folium x³ + y³ = 3xy loops back over itself, so no formula y = f(x) describes it, and yet at almost every point it has a perfectly good tangent. Differentiating the equation as it stands gives the slope, −(∂F/∂x) ÷ (∂F/∂y), and the only points where that fails are the ones where the curve turns vertical or crosses itself — which are exactly the points where it stops being a graph.
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![The slope of x² sin(1/x): discontinuous at nought, and never jumping. The derivative of x^2 sin(1/x) on [−0.12, 0.12], with the value at nought marked and the level 0.5 crossed 77 times.](/f/the-slope-of-x2-sin-1-x-discontinuous-at-nought-and-never-60857308.svg)
A slope can swing but never jump
A function can have a slope at every point without that slope changing continuously: x² sin(1/x) has slope nought at the origin and a slope that swings between −1 and 1 however close to the origin it is taken. What a slope cannot do is jump. Darboux proved in 1875 that a derivative takes every value between any two of its values, so a step is never a derivative — and the only way a slope can be discontinuous is by oscillating.
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A slope without a subtraction
The definition of a slope subtracts two nearly equal numbers and divides by a tiny one, and a computer that follows it loses half its digits however carefully the step is chosen. Two pieces of arithmetic avoid the subtraction altogether. Step sideways into the complex numbers by 10⁻²⁰⁰ and read off the imaginary part; or calculate with numbers a + bε in which ε² = 0, and the slope rides along exactly. Run backwards, the second gives all thousand slopes of a function of a thousand variables for under five times the cost of the function.
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Running a computation backwards with little memory
The reverse sweep that gives a whole gradient for the price of a few evaluations needs the computation's intermediate states in reverse order, and storing them all can cost more memory than the machine has. Keep only a few and recompute the rest, and the cheapest schedule has an exact answer found by Andreas Griewank in 1992: with c stored states and no step run more than r times, the longest chain that can be reversed is the binomial coefficient C(c + r, c). Pascal's triangle is the table of what memory buys.