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 important to 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 a function f(x) with respect to x.


We are dividing two numbers infinitely close to 0, and arriving at a finite answer. D is another operator that can be thought 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:


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,


The dx is in fact a differential element. Let’s start with a derivative that we 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 is the 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:


The length is the sum of all the elements, ds, of length. If we have a function f(x), the length element is usually written as ds=1+[df(x)dx]2dx. If we modify this a bit, we get ds=[dx]2+[df(x)]2. Graphically, we could say that the length element is the hypotenuseMathworldPlanetmath of a right triangleMathworldPlanetmath with one leg 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 a derivative. Compare to the prime notation.


A second derivative is represented as follows:


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

Title Leibniz notation
Canonical name LeibnizNotation
Date of creation 2013-03-22 12:30:47
Last modified on 2013-03-22 12:30:47
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 6
Author mathcam (2727)
Entry type Topic
Classification msc 26A24
Related topic Derivative
Related topic FixedPointsOfNormalFunctions
Related topic Differential