# 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 $x$.

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

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,

$$F(x)=\int f(x)\mathit{d}x$$ |

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

$\frac{dF(x)}{dx}$ | $=$ | $f(x)$ | ||

$dF(x)$ | $=$ | $f(x)dx$ | ||

$\int \mathit{d}F(x)$ | $=$ | $\int f(x)\mathit{d}x$ | ||

$F(x)$ | $=$ | $\int f(x)\mathit{d}x$ |

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:

$$s=\int \mathit{d}s$$ |

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.

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

A second derivative is represented as follows:

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

The other derivatives follow as can be expected: $\frac{{d}^{3}y}{d{x}^{3}}$, 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 $\int \frac{{d}^{2}y}{d{x}^{2}}\mathit{d}x$ or $\int \mathit{d}\frac{dy}{dx}$. Either way, we get $\frac{dy}{dx}$.

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 |