differential calculus

differential calculus

differential calculus: see calculus.

Mathematical process of finding the derivative of a function. Defined abstractly as a process involving limits, in practice it may be done using algebraic manipulations that rely on three basic formulas and four rules of operation. The formulas are: (1) the derivative of math.xmath.n is math.nmath.xmath.n − 1, (2) the derivative of sin math.x is cos math.x, and (3) the derivative of the exponential function math.emath.x is itself. The rules are: (1) (math.amath.f + math.bmath.g)' = math.amath.f' + math.bmath.g', (2) (math.fmath.g)' = math.fmath.g' + math.gmath.f', (3) (math.f/math.g)' = (math.gmath.f' − math.fmath.g')/math.g2, and (4) (math.f(math.g))' = math.f'(math.g)math.g', where math.a and math.b are constants, math.f and math.g are functions, and a prime (') indicates the derivative. The last formula is called the chain rule. The derivation and exploration of these formulas and rules is the subject of differential calculus. Seealso integration.

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In mathematics, and more specifically, in differential calculus, the term differential has several interrelated meanings.

Basic notions

Differential geometry

The notion of a differential motivates several concepts in differential geometry (and differential topology).

Algebraic geometry

Differentials are also important in algebraic geometry, and there are several important notions.

Other meanings

The term differential has also been adopted in homological algebra and algebraic topology, because of the role the exterior derivative plays in de Rham cohomology: in a cochain complex (C_bullet, d_bullet), the maps (or coboundary operators) di are often called differentials. Dually, the boundary operators in a chain complex are sometimes called codifferentials.

The properties of the differential also motivate the algebraic notions of a derivation and a differential algebra.

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