Calculus is the branch of mathematics that deals with continuous change, built from two operations that turn out to be inverses of one another. Differentiation finds the instantaneous rate at which a quantity is changing, the exact speed of a car at one moment rather than its average speed over a trip, or the exact slope of a curve at a single point rather than the slope of a straight line connecting two distant points on it. Integration does the reverse, reconstructing a total quantity, such as the distance a car has traveled, from a complete record of how fast it was going at every moment along the way.
Both operations rest on the idea of a limit: what happens to a calculation as some quantity, an interval of time, a segment of a curve, is allowed to shrink toward zero without ever quite reaching it. Ancient mathematicians, including Archimedes, had used limiting arguments to calculate areas and volumes centuries earlier, but it took until the late seventeenth century for the general, systematic machinery of calculus, applicable to essentially any smooth curve or changing quantity, to be worked out, independently, by Isaac Newton and Gottfried Wilhelm Leibniz.
The fundamental theorem of calculus, which formally connects differentiation and integration as inverse operations, is the result that ties the whole subject together, and it is difficult to overstate how much of later physics, engineering, and applied mathematics depends on it. Newton's laws of motion are differential equations, describing how a planet's position changes because of the forces acting on it, or how a bridge bends under load, or how a population grows over time, all require exactly the machinery calculus provides.