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Infinitesimal is a term used in mathematics to describe an infinitely small quantity. The concept was first introduced by the ancient Greek mathematician Archimedes, who used it to approximate the area under a curve. However, it was not until the 17th century that the modern understanding of infinitesimals was developed by Sir Isaac Newton and Gottfried Wilhelm Leibniz.

Infinitesimals are used in calculus to describe the rate of change of a function at a specific point. This is done by considering the function's value at an infinitesimally close point and comparing it to the value at the original point. The difference between these two values is then divided by the infinitesimal distance between the two points, resulting in the derivative of the function at that point.

Infinitesimals are also used in non-standard analysis, a branch of mathematics that extends the real numbers to include infinitesimals and their reciprocals, which are infinitely large. This allows for a more intuitive understanding of calculus and other areas of mathematics.

However, the use of infinitesimals has been controversial in the mathematical community. Some mathematicians argue that infinitesimals are not well-defined and can lead to paradoxes. Others argue that infinitesimals are a useful tool for understanding and calculating with functions.

In conclusion, infinitesimals are a fundamental concept in mathematics, used to describe infinitely small quantities and the rate of change of functions. While their use has been controversial, they remain an important tool in calculus and other areas of mathematics.