Difference quotient

The difference quotient is a term from mathematics. It describes the ratio of the change in size to the change to another, of which the first dependent. In analysis using the difference quotient, to define the derivative of a function. In the numerical analysis, they can be used for solving differential equations, and for the approximate determination of the derivative of a function.

• 4.1 forward difference quotient
• 4.2 backward difference quotient
• 4.3 Central difference quotient

Definition

Is a real-valued function which is defined in the field, and is so called the quotient

Difference quotient of the interval.

If you write and then there is the alternative spelling

Substituting, therefore, we obtain the notation

Geometrically, the difference quotient of the slope of the secant of the graph of through the points and.

Differential calculus

Difference quotients, together with the limit concept, the theoretical basis of the differential calculus. The threshold value of the difference quotient is referred to as a differential quotient or derivative of the function at the point ( short ), provided that the limit exists. The calculation of this limit is called derivation or differentiation. The table shows the derivatives of some functions. Here, the difference quotient is true only for.

Numerical Mathematics

For differentiable functions of the difference quotient can be used as an approximation for the local discharge. In the finite difference method this property for solving differential equations will be used. This is also used for the numerical differentiation of functions.

Here, the difference quotient is not limited to the first derivative. There are difference quotients for higher partial derivatives as well.

Example

It should be.

The graph of a normal parabola. If we want, for example, the derivation approximately calculated in the vicinity of the site, we would choose for a small value, eg 0.001. This gives the value as a difference quotient in the interval. This is the function of the graph in Sekantensteigung interval and an approximation of the slope of the tangent at the point.

Variants

In practice, different variants of the difference quotient can be used the in the definition of, for instance to improve the accuracy in the determination of the local growth, such as to improve the Sekantensteigung of a graph, or to detect the edge of a function whose Sekantensteigung "backwards" towards the interior of its domain.

Forward difference quotient

The above-defined term

Is also called the forward difference quotient, because the determination of the second function value which is necessary to form, by out to the right, ie the "forward" gone.

Backward difference quotient

Analog refers to the expression

As the backward difference quotient as the difference from off to the left, ie "backwards" went to get the second function value.

The central difference quotient

Commonly used is also the central difference quotient, the one example obtained by averaging the forward differences and backward difference quotients. He is by

Given. For him, the sites used for generating the difference lie symmetrically around the value for which the derivative is to be approximated.

In contrast to the previous two difference quotient, the error terms are as approaching the first derivative at the position only by the class, if the function is differentiated twice, the error of the central differential quotient is, if the function is also three times differentiable. For notation see Landau symbols.

Higher difference quotient

Just as the first derivative can be approximated by difference quotients, this is also true for higher order derivatives which are approximated through higher order difference quotient.

For the second derivative can be, for example, the relationship

Be used four times differentiability of the function provided. The standing behind the notation constant can be of dependent.

• Analysis