—differentially, adv./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.b. See 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]
* * *IIn 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 Δy/Δx, 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′(x0)Δx (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.
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