Calculus
Introduction
Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.
It has two major branches, differential calculus and integral calculus; the former concerns instantaneous rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. --Wikipedia 2023
Limits and Infinitesimals
Calculus is usually developed by working with very small quantities. Historically, the first method of doing so was with infanitesimals. These are objects which can be treated like real numbers but which are, in some sense, "infinitely small". Thus calculus becomes the practice of manipulating infinitesimals.
Later the concept of the limit replaced as it is more useful in most cases. For more information, see Limits.
Differential Calculus
Differential calculus is the study of the derivative of a function. The previously linked document goes into detail. However to summarize, given a function and a point in the domain, the derivative at the point is the slope of the tangent line to the function.
Integral Calculus
Integral Calculus is the study of the accumulation of quantities. The previously linked document goes into detail. It's easy to summarize integrals as the inverse of derivatives.