differential


differential
/dif'euh ren"sheuhl/, adj.
1. of or pertaining to difference or diversity.
2. constituting a difference; distinguishing; distinctive: a differential feature.
3. exhibiting or depending upon a difference or distinction.
4. Physics, Mach. pertaining to or involving the difference of two or more motions, forces, etc.
5. Math. pertaining to or involving a derivative or derivatives.
n.
6. a difference or the amount of difference, as in rate, cost, quantity, degree, or quality, between things that are comparable.
7. Also called differential gear. Mach. an epicyclic train of gears designed to permit two or more shafts to rotate at different speeds, as a set of gears in an automobile permitting the rear wheels to be driven at different speeds when the car is turning.
8. Math.
a. a function of two variables that is obtained from a given function, y = f(x), and that expresses the approximate increment in the given function as the derivative of the function times the increment in the independent variable, written as dy = f'(x)dx.
b. any generalization of this function to higher dimensions.
9. Com.
a. the difference involved in a differential rate.
10. Physics. the quantitative difference between two or more forces, motions, etc.: a pressure differential.
[1640-50; < ML differentialis, equiv. to differenti(a) DIFFERENCE + alis -AL]

* * *

I
In calculus, an expression based on the derivative of a function, useful for approximating certain values of the function.

The differential of an independent variable x, written Δx, is an infinitesimal change in its value. The corresponding differential of its dependent variable y is given by Δy/n=/nf(x/n+/nΔx)/n-/nf(x). Because the derivative of the function f(x), f′(x), is equal to the ratio ΔyΔx as Δx approaches zero (see limit), for small values of Δx, Δy/n≅/nf′(x)Δx. This formula often enables a quick and fairly accurate approximation to be made for what otherwise would be a tedious calculation.
II
(as used in expressions)

* * *

      in mathematics, an expression based on the derivative of a function, useful for approximating certain values of the function. The derivative of a function at the point x0, written as f′(x0), is defined as the limit as Δx approaches 0 of the quotient Δyx, in which Δy is f(x0 + Δx) − f(x0). Because the derivative is defined as the limit, the closer Δx is to 0, the closer will be the quotient to the derivative. Therefore, if Δx is small, then Δy ≈ f′(x0x (the wavy lines mean “is approximately equal to”). For example, to approximate f(17) for f(x) = √x, first note that its derivative f′(x) is equal to (x−1/2)/2. Choosing a computationally convenient value for x0, in this case the perfect square 16, results in a simple calculation of f′(x0) as 1/8 and Δx as 1, giving an approximate value of 1/8 for Δy. Because f(16) is 4, it follows that f(17), or √17, is approximately 4.125, the actual value being 4.123 to three decimal places.

* * *


Universalium. 2010.

Synonyms:

Look at other dictionaries:

  • Differential — may refer to: Contents 1 Mathematics 2 Natural sciences and engineering 3 Social sciences 4 Medicine 5 …   Wikipedia

  • differential — dif fer*en tial, a. [Cf. F. diff[ e]rentiel.] 1. Relating to or indicating a difference; creating a difference; discriminating; special; as, differential characteristics; differential duties; a differential rate. [1913 Webster] For whom he… …   The Collaborative International Dictionary of English

  • Differential — Dif fer*en tial, n. 1. (Math.) An increment, usually an indefinitely small one, which is given to a variable quantity. [1913 Webster] Note: According to the more modern writers upon the differential and integral calculus, if two or more… …   The Collaborative International Dictionary of English

  • differential — is a noun and adjective with several technical and general meanings. As an adjective it is used principally in combinations such as differential rates of interest, which are not just different rates of interest but rates based on quantifiable… …   Modern English usage

  • differential — [dif΄ər en′shəl] adj. [ML differentialis < L differentia: see DIFFERENCE] 1. of, showing, or depending on a difference or differences [differential rates] 2. constituting or making a specific difference; distinguishing [differential qualities] …   English World dictionary

  • Differential — Differential, eine unendlich kleine Aenderung einer stetig veränderlichen Größe x. Das Differential, das mit dx bezeichnet wird, besitzt dieselben Merkmale wie x, aber ein abstufbares Merkmal hat den Wert Null, z.B. Bogendifferential ist ein… …   Lexikon der gesamten Technik

  • differential — I noun attribute, characteristic, constituting a difference, contrasting quality, delicate distinction, difference of degree, differentiating trait, distinction, distinctive feature, distinguishing feature, feature, gradation, grade, graduation,… …   Law dictionary

  • Differential — (v. lat. Differentiale), 1) (Math.), s.u. Differentialrechnung; 2) Chemisches D., so v.w. Atom …   Pierer's Universal-Lexikon

  • Differentiāl — in Verbindung mit einem Hauptwort bei Maschinen oder Maschinenteilen, Apparaten etc. oft angewendet (z. B. Differentialflaschenzug, Differentialpumpe etc.), um im allgemeinen die Wirkung zweier, in entgegengesetztem Sinne tätigen Einflüsse zu …   Meyers Großes Konversations-Lexikon

  • Differential... — Differentiāl... (neulat.) bedeutet bei Bezeichnungen maschineller Vorrichtungen, daß eine Hauptbewegung durch eine entgegengesetzte Bewegung geschwächt wird, so daß nur die Differenz beider Bewegungen zum Ausdruck kommt, z.B. Differentialgetriebe …   Kleines Konversations-Lexikon

  • differential... — differential...,   Schreibvariante für differenzial …   Universal-Lexikon


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.