# Leibniz notation

Leibniz notation centers around the concept of a differential  element  . The differential element of $x$ is represented by $dx$. You might think of $dx$ as being an infinitesimal   change in $x$. It is important to note that $d$ is an operator, not a variable. So, when you see $\frac{dy}{dx}$, you can’t automatically write as a replacement $\frac{y}{x}$.

We use $\frac{df(x)}{dx}$ or $\frac{d}{dx}f(x)$ to represent the derivative   of a function $f(x)$ with respect to $x$.

 $\frac{df(x)}{dx}=\lim_{Dx\to 0}\frac{f(x+Dx)-f(x)}{Dx}$

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

 $\frac{dy}{dx}=\frac{dy}{dx}\frac{du}{du}=\frac{dy}{du}\frac{du}{dx}$

The two $du$s can be cancelled out to arrive at the original derivative. This is the Leibniz notation for the Chain Rule  .

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

 $F(x)=\int f(x)dx$

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

 $\displaystyle\frac{dF(x)}{dx}$ $\displaystyle=$ $\displaystyle f(x)$ $\displaystyle dF(x)$ $\displaystyle=$ $\displaystyle f(x)dx$ $\displaystyle\int dF(x)$ $\displaystyle=$ $\displaystyle\int f(x)dx$ $\displaystyle F(x)$ $\displaystyle=$ $\displaystyle\int f(x)dx$

We can think of $dF(x)$ as the differential element of area. Since $dF(x)=f(x)dx$, the element of area is a rectangle   , with $f(x)\times dx$ as its dimensions. Integration is the sum of all these infinitely thin elements of area along a certain interval   . 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 formula   is often seen as the following:

 $s=\int ds$

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 $ds=\sqrt{1+[\frac{df(x)}{dx}]^{2}}dx$. If we modify this a bit, we get $ds=\sqrt{[dx]^{2}+[df(x)]^{2}}$. Graphically, we could say that the length element is the hypotenuse  of a right triangle  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.

 $f^{\prime}(a)=\left.\frac{df(x)}{dx}\right|_{x=a}$

A second derivative is represented as follows:

 $\frac{d}{dx}\frac{dy}{dx}=\frac{d^{2}y}{dx^{2}}$

The other derivatives follow as can be expected: $\frac{d^{3}y}{dx^{3}}$, 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 $\int\frac{d^{2}y}{dx}$? We could turn it into $\int\frac{d^{2}y}{dx^{2}}dx$ or $\int d\frac{dy}{dx}$. Either way, we get $\frac{dy}{dx}$.

Title Leibniz notation LeibnizNotation 2013-03-22 12:30:47 2013-03-22 12:30:47 mathcam (2727) mathcam (2727) 6 mathcam (2727) Topic msc 26A24 Derivative FixedPointsOfNormalFunctions Differential