differential

differential

If a

real function

f

has the

derivative

at a value

x

of its

argument

,
then the

absolute value

of the expression

f

⁢

(

x

+

Δ

⁢

x

)

-

f

⁢

(

x

)

Δ

⁢

x

-

f

′

⁢

(

x

)

may be made smaller than any given positive number by making

|

Δ

⁢

x

|

sufficiently small.  If we generally denote by

⟨

Δ

⁢

x

⟩

an expression having such a property, we can write

f

⁢

(

x

+

Δ

⁢

x

)

-

f

⁢

(

x

)

Δ

⁢

x

-

f

′

⁢

(

x

)

=

⟨

Δ

⁢

x

⟩

.

This allows us to express the increment of the

function

f

⁢

(

x

+

Δ

⁢

x

)

-

f

⁢

(

x

)

:=

Δ

⁢

f

in the form

Δ

⁢

f

=

(

f

′

⁢

(

x

)

+

⟨

Δ

⁢

x

⟩

)

⁢

Δ

⁢

x

=

f

′

⁢

(

x

)

⁢

Δ

⁢

x

+

⟨

Δ

⁢

x

⟩

⁢

Δ

⁢

x

.

(1)

This result may be uttered as the

Theorem

.

If the derivative

f

′

⁢

(

x

)

exists, then the increment

Δ

⁢

f

of the function corresponding to the increment of the argument from

x

to

x

+

Δ

⁢

x

may be divided into two essentially different parts:

1

∘

. One part is proportional to the increment

Δ

⁢

x

of the argument, i.e. it equals this increment multiplied by a

coefficient

f

′

⁢

(

x

)

which is on the increment.

2

∘

. The ratio of the other part

⟨

Δ

⁢

x

⟩

⁢

Δ

⁢

x

to the increment

Δ

⁢

x

of the argument tends to 0 along with

Δ

⁢

x

.

As well, the

converse

of the theorem is true.

By Leibniz, the former part

f

′

⁢

(

x

)

⁢

Δ

⁢

x

is called the

differential

, or the

differential increment

of the function, and denoted by

d

⁢

f

⁢

(

x

)

, briefly

d

⁢

f

.

It is easily checked that when one has set the

tangent line

of the curve at the point

(

x

,

f

⁢

(

x

)

)

,  the differential increment

d

⁢

f

⁢

(

x

)

geometrically means the increment of the

ordinate

of the corresponding to the transition from the

abscissa

x

to the ascissa

x

+

Δ

⁢

x

.

The differential of the

identity function

(

f

⁢

(

x

)

≡

x

,

f

′

⁢

(

x

)

≡

1

) is

d

⁢

x

=

1

⋅

Δ

⁢

x

=

Δ

⁢

x

.

Accordingly, one can without discrepancies denote the increment

Δ

⁢

x

of the variable

x

by

d

⁢

x

.  Therefore, the differential of a function

f

gets the notation

d

⁢

f

⁢

(

x

)

=

f

′

⁢

(

x

)

⁢

d

⁢

x

.

(2)

It follows

f

′

⁢

(

x

)

=

d

⁢

f

⁢

(

x

)

d

⁢

x

,

(3)

in which the

differential

quotient

is often replaced using a “

differentiation

operator

”:

f

′

⁢

(

x

)

=

d

d

⁢

x

⁢

f

⁢

(

x

)

(4)

Remark.

One can write certain for forming differentials.  For example, if

f

and

g

are two

differentiable functions

, one has

d

⁢

(

f

+

g

)

=

d

⁢

f

+

d

⁢

g

,

d

⁢

(

f

⁢

g

)

=

f

⁢

d

⁢

g

+

g

⁢

d

⁢

f

.

(5)

Naturally, they seem

trivial

consequences

of the

sum rule

and the

product rule

, but they include a deeper contents in the case where

f

and

g

depend on more than one variable (see

total differential

(

http://planetmath.org/TotalDifferential

)).

As for a

composite function

, e.g.

h

=

f

∘

u

, the

chain rule

and (2) yield

d

⁢

f

⁢

(

u

⁢

(

x

)

)

=

f

′

⁢

(

u

⁢

(

x

)

)

⁢

u

′

⁢

(

x

)

⁢

d

⁢

x

=

f

′

⁢

(

u

⁢

(

x

)

)

⁢

d

⁢

u

⁢

(

x

)

,

i.e. simply

d

⁢

h

=

f

′

⁢

(

u

)

⁢

d

⁢

u

.

References

1

Ernst Lindelöf

:

Johdatus korkeampaan analyysiin

. Fourth edition. Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).

2

E. Lindelöf

:

Einführung in die höhere

Analysis

. Nach der ersten schwedischen
und zweiten finnischen Auflage auf deutsch herausgegeben von E. Ullrich. Teubner, Leipzig (1934).

Title

differential

Canonical name

Differential

Date of creation

2014-02-23 15:34:46

Last modified on

2014-02-23 15:34:46

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

23

Author

pahio (2872)

Entry type

Definition

Classification

msc 53A04

Classification

msc 26A06

Classification

msc 26-03

Classification

msc 01A45

Synonym

differential increment

Related topic

Derivative2

Related topic

LeibnizNotation

Related topic

ExactDifferentialEquation

Related topic

ProductAndQuotientOfFunctionsSum

Related topic

TotalDifferential

Defines

differential quotient

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