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The application of hyperreal numbers to the foundations of calculus is called nonstandard analysis. This provides a way to define the basic concepts of calculus such as the derivative and integral in terms of infinitesimals, thereby giving a precise meaning to the in the Leibniz notation.
Calculus is of vital importance in physics: many physical processes are described by equations involving derivatives, called differential equations. Physics is particularly concerned with the way quantities change and develop over time, and the concept of the " time derivative " — the rate of change over time — is essential for the precise ...
There are many alternatives to the classical calculus of Newton and Leibniz; for example, each of the infinitely many non-Newtonian calculi. [1] Occasionally an alternative calculus is more suited than the classical calculus for expressing a given scientific or mathematical idea. [2] [3] [4]
The derivative f′(x) of a curve at a point is the slope (rise over run) of the line tangent to that curve at that point. Differential calculus is the study of the definition, properties, and applications of the derivative of a function. The process of finding the derivative is called differentiation. Given a function and a point in the domain ...
The corresponding derivative is calculated using Lagrange's rule for differential operators. To find the α th order derivative, the n th order derivative of the integral of order (n − α) is computed, where n is the smallest integer greater than α (that is, n = ⌈α⌉). The Riemann–Liouville fractional derivative and integral has ...
The derivatives in the table above are for when the range of the inverse secant is [,] and when the range of the inverse cosecant is [,]. It is common to additionally define an inverse tangent function with two arguments , arctan ( y , x ) . {\displaystyle \arctan(y,x).}
In calculus, the differential represents the principal part of the change in a function = with respect to changes in the independent variable. The differential is defined by = ′ (), where ′ is the derivative of f with respect to , and is an additional real variable (so that is a function of and ).
The calculus has been applied to stochastic partial differential equations as well. The calculus allows integration by parts with random variables; this operation is used in mathematical finance to compute the sensitivities of financial derivatives. The calculus has applications in, for example, stochastic filtering.