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Geometrically, this difference quotient measures the slope of the secant line passing through the points with coordinates (a, f(a)) and (b, f(b)). [10] Difference quotients are used as approximations in numerical differentiation, [8] but they have also been subject of criticism in this application. [11]
For differentiable functions, the symmetric difference quotient does provide a better numerical approximation of the derivative than the usual difference quotient. [3] The symmetric derivative at a given point equals the arithmetic mean of the left and right derivatives at that point, if the latter two both exist. [1] [2]: 6
In mathematical analysis, Schwarz's theorem (or Clairaut's theorem on equality of mixed partials) [9] named after Alexis Clairaut and Hermann Schwarz, states that for a function : defined on a set , if is a point such that some neighborhood of is contained in and has continuous second partial derivatives on that neighborhood of , then for all i ...
The simplest method is to use finite difference approximations. A simple two-point estimation is to compute the slope of a nearby secant line through the points (x, f(x)) and (x + h, f(x + h)). [1] Choosing a small number h, h represents a small change in x, and it can be either positive or negative.
When this happens, the limit of the product of these two factors will equal the product of the limits of the factors. The two factors are Q(g(x)) and (g(x) − g(a)) / (x − a). The latter is the difference quotient for g at a, and because g is differentiable at a by assumption, its limit as x tends to a exists and equals g′(a).
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For example, consider the ordinary differential equation ′ = + The Euler method for solving this equation uses the finite difference quotient (+) ′ to approximate the differential equation by first substituting it for u'(x) then applying a little algebra (multiplying both sides by h, and then adding u(x) to both sides) to get (+) + (() +).
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