Leibniz notation

Leibniz notation

Leibniz notation

centers around the

concept

of a

differential

element

.
The differential element of

x

is represented by

d

⁢

x

.
You might think of

d

⁢

x

as being an

infinitesimal

change in

x

. It is important
to note that

d

is an

operator

, not a variable. So, when you see

d

⁢

y

d

⁢

x

,
you can’t automatically write as a replacement

y

x

.

We use

d

⁢

f

⁢

(

x

)

d

⁢

x

or

d

d

⁢

x

⁢

f

⁢

(

x

)

to represent the

derivative

of a

function

f

⁢

(

x

)

with respect to

x

.

d

⁢

f

⁢

(

x

)

d

⁢

x

=

lim

D

⁢

x

→

0

⁡

f

⁢

(

x

+

D

⁢

x

)

-

f

⁢

(

x

)

D

⁢

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

D

⁢

x

as

D

⁢

x

approaches 0,
we get an infinitesimal change

d

⁢

x

.

Leibniz notation shows a wonderful use in the following example:

d

⁢

y

d

⁢

x

=

d

⁢

y

d

⁢

x

⁢

d

⁢

u

d

⁢

u

=

d

⁢

y

d

⁢

u

⁢

d

⁢

u

d

⁢

x

The two

d

⁢

u

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

)

=

∫

f

⁢

(

x

)

⁢

𝑑

x

The

d

⁢

x

is in fact a differential element. Let’s start with a derivative that
we know (since

F

⁢

(

x

)

is an

antiderivative

of

f

⁢

(

x

)

).

d

⁢

F

⁢

(

x

)

d

⁢

x

=

f

⁢

(

x

)

d

⁢

F

⁢

(

x

)

=

f

⁢

(

x

)

⁢

d

⁢

x

∫

𝑑

F

⁢

(

x

)

=

∫

f

⁢

(

x

)

⁢

𝑑

x

F

⁢

(

x

)

=

∫

f

⁢

(

x

)

⁢

𝑑

x

We can think of

d

⁢

F

⁢

(

x

)

as the differential element of area. Since

d

⁢

F

⁢

(

x

)

=

f

⁢

(

x

)

⁢

d

⁢

x

,
the element of area is a

rectangle

, with

f

⁢

(

x

)

×

d

⁢

x

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

=

∫

𝑑

s

The length is the sum of all the elements,

d

⁢

s

, of length. If we have a function

f

⁢

(

x

)

, the length element is usually written as

d

⁢

s

=

1

+

[

d

⁢

f

⁢

(

x

)

d

⁢

x

]

2

⁢

d

⁢

x

. If we
modify this a bit, we get

d

⁢

s

=

[

d

⁢

x

]

2

+

[

d

⁢

f

⁢

(

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

′

⁢

(

a

)

=

d

⁢

f

⁢

(

x

)

d

⁢

x

|

x

=

a

A

second derivative

is represented as follows:

d

d

⁢

x

⁢

d

⁢

y

d

⁢

x

=

d

2

⁢

y

d

⁢

x

2

The other derivatives follow as can be expected:

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

∫

d

2

⁢

y

d

⁢

x

? We
could turn it into

∫

d

2

⁢

y

d

⁢

x

2

⁢

𝑑

x

or

∫

𝑑

d

⁢

y

d

⁢

x

.
Either way, we get

d

⁢

y

d

⁢

x

.

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

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