It is used in almost every field of science and much more! Here is a link to something fun that uses calculus. http://www.geom.uiuc.edu/apps/rainbow/
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The difference between Leibniz calculus to Newton calculus was that Leibniz developed Newton's calculus into the calculus we all know today. For instance, diffentiation and intergration, limits, continuity, etc. This type of calculus was the pure mathematics. On the otherhand, the calculus which Newton found was that used in physics, such as speed and velocity which helped with physics greatly. Today, calculus not only used in just mathematics or physics, but used in finance, as well as exploited in engineering.
In the 'real world', the purpose of a course of study in pre-calculus is to prepare the student for a course of study in Calculus.
For Literally Everything.
The term calculus comes directly from Latin. In Latin a calculus (noun) is a small stone used for counting, much like the beads on an abacus. One of the fundamental uses for modern calculus is integration, which is of course addition of infinitely small sections.
Newton is the named founder of Calculus. Yet there is controversy because it is claimed that Leibniz stole Newton's Calculus notes and took all credit for Calculus. But to this day Leibniz's integral and derivative notation is more commonly used that Newton's which was found confusing.