What is the difference quotient

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Difference Quotient

what is the difference quotient

In single-variable calculus, the difference quotient is usually the name for the expression. f (x + h) ? f (x) h {\displaystyle {\frac {f(x+h)-f(x)}{h}}} {\frac {f(x+h)-f(x) }.

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The difference quotient of a function between two distinct points in its domain is defined as the quotient of the difference between the function values at the two points by the difference between the two points. In symbols, if is a function defined on some subset of the reals and are distinct elements in the domain of , then the difference quotient of between and , denoted , is defined as:. Note that the definition is symmetric in and , i. The difference quotient of a function between two distinct points in its domain is defined as the slope of the chord joining the corresponding points in the graph of the function. In symbols, if is a function defined on some subset of the reals and are distinct elements in the domain of , then the difference quotient of between and is defined as the slope of the line segment joining the points and , both of which are part of the graph of.

Before we define the difference quotient and the difference quotient formula, it is important to first understand the definition of derivatives. In the most basic sense, a derivative is a measure of a function's rate of change. That is, the instantaneous rate of change at a given point. In order to measure this value, we use what is called a tangent line, and measure its slope to given the best possible linear approximation at a single point. In this article, however, we won't be looking at derivatives in particular. Instead, we will be looking at the difference quotient, which is a stepping stone to calculating derivatives of functions.

One of the cornerstones of calculus is the difference quotient. The difference quotient along with limits allows you to take the regular old slope formula that you used to compute the slope of lines in algebra class and use it for the calculus task of calculating the slope or derivative of a curve. To compute the slope, you need two points to plug into this formula. For a line, this is easy. You just pick any two points on the line and plug them in.

In single-variable calculus , the difference quotient is usually the name for the expression. By a slight change in notation and viewpoint , for an interval [ a , b ], the difference quotient. Difference quotients are used as approximations in numerical differentiation , [8] but they have also been subject of criticism in this application. The difference quotient is sometimes also called the Newton quotient [10] [12] [13] [14] after Isaac Newton or Fermat's difference quotient after Pierre de Fermat. The typical notion of the difference quotient discussed above is a particular case of a more general concept.



The Difference Quotient (Illustrated)

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line through two points on the graph of f. These are the points with x- coordinates x and x + h. The difference quotient is used in the definition the derivative.
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