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In mathematics, divided differences is an algorithm, historically used for computing tables of logarithms and trigonometric functions. [citation needed] Charles Babbage's difference engine, an early mechanical calculator, was designed to use this algorithm in its operation. [1] Divided differences is a recursive division process.
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let () = (), where both f and g are differentiable and ()
This formula can be obtained by Taylor series expansion: (+) = + ′ ()! ″ ()! () +. The complex-step derivative formula is only valid for calculating first-order derivatives. A generalization of the above for calculating derivatives of any order employs multicomplex numbers , resulting in multicomplex derivatives.
The differential was first introduced via an intuitive or heuristic definition by Isaac Newton and furthered by Gottfried Leibniz, who thought of the differential dy as an infinitely small (or infinitesimal) change in the value y of the function, corresponding to an infinitely small change dx in the function's argument x.
Differential of a function – Notion in calculus Differentiation of integrals – Problem in mathematics Differentiation under the integral sign – Differentiation under the integral sign formula Pages displaying short descriptions of redirect targets
In mathematical analysis, the mean value theorem for divided differences generalizes the mean value theorem to higher derivatives. [1] Statement of the theorem
A divided difference, however, does require further elucidation, as it equals the average derivative between and including LB and UB: P ( t n ) = L B + T N − 1 U T − 1 Δ B = U B − U T − T N U T − 1 Δ B ; .
The term differential is used nonrigorously in calculus to refer to an infinitesimal ("infinitely small") change in some varying quantity. For example, if x is a variable, then a change in the value of x is often denoted Δx (pronounced delta x). The differential dx represents an infinitely small change in the variable x. The idea of an ...
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