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单词 LeibnizNotation
释义

Leibniz notation


Leibniz notation centers around the concept of a differentialMathworldPlanetmath elementMathworldMathworld.The differential element of x is represented by dx.You might think of dx as being an infinitesimalMathworldPlanetmathPlanetmath change in x. It is importantto note that d is an operator, not a variable. So, when you see dydx,you can’t automatically write as a replacement yx.

We use df(x)dx or ddxf(x) to represent the derivativeMathworldPlanetmathPlanetmath of afunction f(x) with respect to x.

df(x)dx=limDx0f(x+Dx)-f(x)Dx

We are dividing two numbers infinitely close to 0,and arriving at a finite answer. D is another operator that can bethought of just a change in x. When we take the limit of Dx as Dx approaches 0,we get an infinitesimal change dx.

Leibniz notation shows a wonderful use in the following example:

dydx=dydxdudu=dydududx

The two dus can be cancelled out to arrive at the original derivative.This is the Leibniz notation for the Chain RuleMathworldPlanetmath.

Leibniz notation shows up in the most common way of representing an integral,

F(x)=f(x)𝑑x

The dx is in fact a differential element. Let’s start with a derivative thatwe know (since F(x) is an antiderivative of f(x)).

dF(x)dx=f(x)
dF(x)=f(x)dx
𝑑F(x)=f(x)𝑑x
F(x)=f(x)𝑑x

We can think of dF(x) as the differential element of area. Since dF(x)=f(x)dx,the element of area is a rectangleMathworldPlanetmathPlanetmath, with f(x)×dx as its dimensions. Integration isthe sum of all these infinitely thin elements of area along a certain intervalMathworldPlanetmathPlanetmath. The result: a finite number.

(a diagram is deserved here)

One clear advantage of this notation is seen when finding the length s of a curve.The formulaMathworldPlanetmathPlanetmath is often seen as the following:

s=𝑑s

The length is the sum of all the elements, ds, of length. If we have a functionf(x), the length element is usually written as ds=1+[df(x)dx]2dx. If wemodify this a bit, we get ds=[dx]2+[df(x)]2. Graphically, wecould say that the length element is the hypotenuseMathworldPlanetmath of a right triangleMathworldPlanetmath with oneleg being the x element, and the other leg being the f(x) element.

(another diagram would be nice!)

There are a few caveats, such as if you want to take the value of aderivative. Compare to the prime notation.

f(a)=df(x)dx|x=a

A second derivative is represented as follows:

ddxdydx=d2ydx2

The other derivatives follow as can be expected: d3ydx3, etc.You might think this is a little sneaky, but it is the notation. Properly usingthese terms can be interesting. For example, what is d2ydx? Wecould turn it into d2ydx2𝑑x or 𝑑dydx.Either way, we get dydx.

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更新时间:2025/5/4 6:23:26