The first, differential calculus, is concerned with finding the instantaneous rate of change (or derivative) of a function's value with respect to changes in its argument (roughly speaking, how much the value of a function changes with a small change in its argument). This derivative can also be interpreted as the slope of the function's graph at a specific point.
Initially, the derivative is defined via a limit process involving secant slopes. These messy calculations can be avoided however because of powerful differentiation rules which allow to find derivatives easily using simple algebraic manipulations.
At a maximal or minimal point, a function's derivative must be zero, and this yields a very useful optimization method.
The second branch of calculus, integral calculus, studies methods for finding the integral? of a function. An integral may be defined as the limit of a sum of terms which correspond to areas under the graph of a function. Considered as such, integration allows us to calculate the area under a curve and the surface area and volume of solids such as spheres and cones.
The Fundamental Theorem of Calculus states that derivatives and intergrals are inverse operations. This allows to recover the total change in a function over some interval from its instantaneous rate of change, by integrating the latter. The fundamental theorem also provides a method to compute many integrals algebraically, without actually performing the limit process, by finding antiderivatives. It also allows to solve some differential equations, equations that relate an unknown function to its derivative. Differential equations are ubiquitous in the sciences.
The conceptual foundations of calculus include the function, limit, infinite sequences, infinite series and continuity. Its tools include the [Symbol Manipulation]? techniques associated with [School Algebra]?, and mathematical induction.
Calculus has been extended to differential equations, vector calculus, and differential topology. The modern, formally correct version of calculus is known as [real analysis]?.
Gottfried Wilhelm Leibniz and Sir Isaac Newton independently invented differential and integral calculus in the late 1600's. Newton (who represented derivatives as f', f'', etc.) provided a host of applications in physics, but Leibniz' superior notation (df/dx, d^2f/(dx)^2, etc.) was eventually adopted.
Other disciplines called "calculus" include
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