derivative
A derivative measures how fast something is changing at a single instant. Picture driving a car: your position keeps shifting, and the derivative of that position is exactly your speedometer reading — how quickly your location is changing right now, this very moment, not averaged over the whole trip.
Geometrically, it's the steepness of a curve at one point. Imagine zooming in on a winding graph until the curve looks perfectly straight; the slope of that tiny straight piece is the derivative there. A steep climb means a large derivative; a flat stretch means a derivative of zero, the curve momentarily level.
The clever trick at its heart is taking an average rate of change over a tinier and tinier window — and seeing what number it homes in on as the window shrinks toward nothing. That limit is the derivative. A common confusion: it doesn't tell you how big something is, only how fast it's changing — a car can sit at 100 km on the odometer yet have a speed (derivative) of zero while parked.
The derivative as a limit: the average change over an interval h, as h shrinks to zero.
Newton and Leibniz invented calculus independently in the late 1600s and quarreled bitterly over the credit. Leibniz's notation dy/dx — evoking an infinitely small change in y divided by an infinitely small change in x — won out, and we still write derivatives his way today.