derivative

A derivative is the mathematics of change. If a quantity y depends on another quantity x, the derivative dy/dx tells you how fast y changes when x changes by a tiny amount. In physics, the most important derivative is the time derivative: velocity is the derivative of position, acceleration is the derivative of velocity, and force — through F = ma — reaches into the second derivative of position itself.

Geometrically, a derivative is a slope. Draw a curve. Pick a point on it. Now pick a nearby point and draw the straight line through both — that is a secant, and its slope is the average rate of change between them. Now slide the second point closer and closer to the first. The secant rotates, and in the limit where the two points merge, it becomes the tangent line — the line that just touches the curve at that single point. The slope of that tangent is the derivative. Nicole Oresme, in the fourteenth century, used exactly this area-under-the-velocity-curve idea to derive what we now call the mean-speed theorem, long before calculus existed as a formal subject.

Newton and Leibniz independently invented the derivative in the 1660s and 1670s, each with their own notation and each with their own priority fight. Newton called it a fluxion; Leibniz wrote it dy/dx, and his notation won. The derivative is the first half of calculus — the other half is the integral, which undoes it. Together, they are the mathematical language in which almost every law of modern physics is written.