Calculus is a branch of mathematics with applications in just about all areas of science, including physics, chemistry, biology, sociology and economics. Calculus was invented in the 17th century independently by two of the greatest mathematicians who ever lived, the English physicist and mathematician Sir Isaac Newton and the German mathematician Gottfried Leibniz. Calculus allows us to perform calculations that would be practically impossible without it.
The most basic concept in calculus is the notion of a derivative. The derivative f'(x) of a function f(x) is a new function whose value is equal to the slope of the original function at every value of the argument x. In order to define the slope of a function at a point, it is first necessary to study infinitely-small quantities known as infinitesimal quantities. This is probably the hardest part of calculus. Once everything, such as the derivative, is defined in terms of these infintesimal quantities, it is relatively easy to work with them.
Another basic concept in calculus is the integral, which is used to compute areas and volumes. Although it is not at all obvious, the integral turns out to be the opposite of the derivative. Still, computing integrals, i.e. integration, is in general more difficult than computing derivatives, i.e. differentiation.
Isaac Newton invented calculus in order to solve certain problems in physics, such as computing the gravitational force on an apple near the surface of the earth. Thus, it should come as little surprise that physics is perhaps the discipline for which calculus is most widely used. However, it is also widely used in other areas of science as well. For instance, calculus is used in chemistry to compute reaction rates, in biology and sociology to model population growth, and in economics to model compound interest.