Calculus, more properly called analysis (or real analysis or, in older literature, infinitesimal analysis), is the branch of mathematics studying the rate of change of quantities (which can be interpreted as slopes of curves) and the length, area, and volume of objects. Calculus is sometimes divided into differential and integral calculus, which are concerned with derivatives $$\frac{d}{dx}f(x) = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$$

and integrals $$\int_a^b f(x) dx = \lim_{\Delta x \to 0} \sum_{k=0}^n f(x_i)\Delta x_i$$


While ideas related to calculus had been known for some time (Archimedes' method of exhaustion was a form of calculus), it was not until the independent work of Newton and Leibniz that the modern elegant tools and ideas of calculus were developed.

Even so, many years elapsed until the subject was put on a mathematically rigorous footing by mathematicians such as Weierstrass, which famously proved an everywhere continous function can be nowhere differentiable.

Source: Wolfram Mathworld

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