differentiation [dif΄əren΄shē ā′shən]n.1. a differentiating or being differentiated2. Biol. the modification of an organ, tissue, etc. in structure or function during development into a more specialized state3. Math. the working out of the differential or derivative
* * *dif·fer·en·ti·a·tion (dĭf'ə-rĕn'shē-āʹshən) n.1.a. The act or process of differentiating.b. The state of becoming differentiated.2. Mathematics. The process of computing a derivative.3. Biology. The process by which cells or tissues undergo a change toward a more specialized form or function, especially during embryonic development.
* * *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 xn is nxn -1, (2) the derivative of sin x is cos x, and (3) the derivative of the exponential function ex is itself. The rules are: (1) (af + bg)′ = af′ + bg′, (2) (fg)′ = fg′ + gf′, (3) (f/g)′ = (gf′ -fg′)/g2, and (4) (f(g))′ = f′(g)g′, where a and b are constants, f and 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. See also integration.
* * *in mathematics, process of finding the derivative, or rate of change, of a function. In contrast to the abstract nature of the theory behind it, the practical technique of differentiation can be carried out by purely algebraic manipulations, using three basic derivatives, four rules of operation, and a knowledge of how to manipulate functions.The three basic derivatives (D) are: (1) for algebraic functions, D(xn) = nxn − 1, in which n is any real number; (2) for trigonometric functions, D(sin x) = cos x; and (3) for exponential functions (exponential function), D(ex) = ex.For functions built up of combinations of these classes of functions, the theory provides the following basic rules for differentiating the sum, product, or quotient of any two functions f(x) and g(x) the derivatives of which are known (where a and b are constants): D(af + bg) = aDf + bDg (sums); D(fg) = fDg + gDf (products); and D(f/g) = (gDf − fDg)/g2 (quotients).The other basic rule, called the chain rule, provides a way to differentiate a composite function. If f(x) and g(x) are two functions, the composite function f(g(x)) is calculated for a value of x by first evaluating g(x) and then evaluating the function f at this value of g(x); for instance, if f(x) = sin x and g(x) = x2, then f(g(x)) = sin x2, while g(f(x)) = (sin x)2. The chain rule states that the derivative of a composite function is given by a product, as D(f(g(x))) = Df(g(x)) ∙ Dg(x). In words, the first factor on the right, Df(g(x)), indicates that the derivative of Df(x) is first found as usual, and then x, wherever it occurs, is replaced by the function g(x). In the example of sin x2, the rule gives the result D(sin x2) = Dsin(x2) ∙ D(x2) = (cos x2) ∙ 2x.In the German mathematician Gottfried Wilhelm Leibniz (Leibniz, Gottfried Wilhelm)'s notation, which uses d/dx in place of D and thus allows differentiation with respect to different variables to be made explicit, the chain rule takes the more memorable “symbolic cancellation” form:d(f(g(x)))/dx = df/dg ∙ dg/dx.
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