derivative

derivative

Qualitatively the

derivative

is a of the change of a

function

in a small around a specified point.

Motivation

The idea behind the derivative comes from the straight line. What
characterizes a straight line is the fact that it has constant
“slope”.

Figure 1:

The straight line

y

=

m

⁢

x

+

b

In other words, for a line given by the equation

y

=

m

⁢

x

+

b

, as in Fig.

1

, the ratio of

Δ

⁢

y

over

Δ

⁢

x

is always constant
and has the value

Δ

⁢

y

Δ

⁢

x

=

m

.

Figure 2:

The

parabola

y

=

x

2

and its

tangent

at

(

x

0

,

y

0

)

For other curves we cannot define a “slope”, like for the straight
line, since such a quantity would not be constant. However, for
sufficiently

smooth curves

, each point on a curve has a tangent line.
For example consider the curve

y

=

x

2

, as in Fig.

2

. At
the point

(

x

0

,

y

0

)

on the curve, we can draw a tangent of slope

m

given by the equation

y

-

y

0

=

m

⁢

(

x

-

x

0

)

.

Suppose we have a curve of the form

y

=

f

⁢

(

x

)

, and at the point

(

x

0

,

f

⁢

(

x

0

)

)

we have a tangent given by

y

-

y

0

=

m

⁢

(

x

-

x

0

)

. Note that
for values of

x

sufficiently close to

x

0

we can make the
approximation

f

⁢

(

x

)

≈

m

⁢

(

x

-

x

0

)

+

y

0

. So the slope

m

of the
tangent describes how much

f

⁢

(

x

)

changes in the

vicinity

of

x

0

. It
is the slope of the tangent that will be associated with the
derivative of the function

f

⁢

(

x

)

.

Formal definition

More formally for any

real function

f

:

ℝ

→

ℝ

, we define the

derivative

of

f

at the point

x

as the following limit (if
it exists)

f

′

⁢

(

x

)

:=

lim

h

→

0

⁡

f

⁢

(

x

+

h

)

-

f

⁢

(

x

)

h

.

This definition turns out to be with
the motivation introduced above.

The derivatives for some

elementary functions

are (cf.

derivative
notation

)

1.

d

d

⁢

x

⁢

c

=

0

, where

c

is constant;

2.

d

d

⁢

x

⁢

x

n

=

n

⁢

x

n

-

1

;

3.

d

d

⁢

x

⁢

sin

⁡

x

=

cos

⁡

x

;

4.

d

d

⁢

x

⁢

cos

⁡

x

=

-

sin

⁡

x

;

5.

d

d

⁢

x

⁢

e

x

=

e

x

;

6.

d

d

⁢

x

⁢

ln

⁡

x

=

1

x

.

While derivatives of more complicated expressions can be
calculated algorithmically using the following rules

Linearity

d

d

⁢

x

⁢

(

a

⁢

f

⁢

(

x

)

+

b

⁢

g

⁢

(

x

)

)

=

a

⁢

f

′

⁢

(

x

)

+

b

⁢

g

′

⁢

(

x

)

;

Product rule

d

d

⁢

x

⁢

(

f

⁢

(

x

)

⁢

g

⁢

(

x

)

)

=

f

′

⁢

(

x

)

⁢

g

⁢

(

x

)

+

f

⁢

(

x

)

⁢

g

′

⁢

(

x

)

;

Chain rule

d

d

⁢

x

⁢

g

⁢

(

f

⁢

(

x

)

)

=

g

′

⁢

(

f

⁢

(

x

)

)

⁢

f

′

⁢

(

x

)

;

Quotient Rule

d

d

⁢

x

⁢

f

⁢

(

x

)

g

⁢

(

x

)

=

f

′

⁢

(

x

)

⁢

g

⁢

(

x

)

-

f

⁢

(

x

)

⁢

g

′

⁢

(

x

)

g

⁢

(

x

)

2

.

Note that the quotient rule, although given as much importance as the
other rules in elementary

calculus

, can be derived by succesively
applying the product rule and the chain rule to

f

⁢

(

x

)

g

⁢

(

x

)

=

f

⁢

(

x

)

⁢

1

g

⁢

(

x

)

. Also the quotient rule does not generalize as well as the other ones.

Since the derivative

f

′

⁢

(

x

)

of

f

⁢

(

x

)

is also a function

x

,
higher derivatives can be obtained by applying the same procedure
to

f

′

⁢

(

x

)

and so on.

Generalization

Banach Spaces

Unfortunately the notion of the “slope of the tangent” does not
directly generalize to more abstract situations. What we can do is
keep in mind the facts that the tangent is a

linear function

and that
it approximates the function near the point of tangency, as well as
the formal definition above.

Very general conditions under which we can define a derivative in
a manner much

similar

to the above areas follows.
Let

f

:

𝖵

→

𝖶

, where

𝖵

and

𝖶

are

Banach
spaces

. Let

𝐡

≠

0

be an

element of

𝖵

. We define the

directional derivative

(

D

𝐡

⁢

f

)

⁢

(

𝐱

)

at

𝐱

as the following
limit (when it exists):

(

D

𝐡

⁢

f

)

⁢

(

𝐱

)

:=

lim

ϵ

→

0

⁡

f

⁢

(

𝐱

+

ϵ

⁢

𝐡

)

-

f

⁢

(

𝐱

)

ϵ

,

where

ϵ

is a scalar. Note that

f

⁢

(

x

+

ϵ

⁢

𝐡

)

≈

f

⁢

(

𝐱

)

+

ϵ

⁢

(

D

𝐡

⁢

f

)

⁢

(

𝐱

)

, which is with
our original motivation. In certain contexts, this directional derivative is also called the

Gâteaux derivative

.

Finally we define the

derivative

at

𝐱

as the

bounded linear map

(

D

⁢

f

)

⁢

(

𝐱

)

:

𝖵

→

𝖶

such that for any
non-zero

𝐡

∈

𝖵

lim

∥

𝐡

∥

→

0

⁡

(

f

⁢

(

𝐱

+

𝐡

)

-

f

⁢

(

𝐱

)

)

-

(

D

⁢

f

)

⁢

(

𝐱

)

⋅

𝐡

∥

𝐡

∥

=

0

.

Once again we have

f

⁢

(

𝐱

+

𝐡

)

≈

f

⁢

(

𝐱

)

+

(

D

⁢

f

)

⁢

(

𝐱

)

⋅

𝐡

. In fact,
if the derivative

(

D

⁢

f

)

⁢

(

𝐱

)

exists, the directional derivatives can
be obtained as

(

D

𝐡

⁢

f

)

⁢

(

𝐱

)

=

(

D

⁢

f

)

⁢

(

𝐱

)

⋅

𝐡

.

1

1

The notation

A

⋅

𝐡

is used when

𝐡

is a
vector and

A

a

linear operator

. This notation can be
considered advantageous to the usual notation

A

⁢

(

𝐡

)

, since
the latter is rather bulky and the former incorporates the
intuitive distributive properties of linear operators also
associated with usual

multiplication

.

However, the
existence of

(

D

𝐡

⁢

f

)

for each non-zero

𝐡

∈

𝖵

does not guarantee
the existence of

(

D

⁢

f

)

⁢

(

𝐱

)

. This derivative is also called the

Fréchet derivative

. In the more familiar case

f

:

ℝ

n

→

ℝ

m

, the derivative

D

⁢

f

is simply the

Jacobian

of

f

.

Under these general conditions the following properties of the
derivative remain

1.

D

⁢

𝐡

=

0

, where

𝐡

is a constant;

2.

D

⁢

(

A

⋅

𝐱

)

=

A

, where

A

is linear.

Linearity

D

⁢

(

a

⁢

f

⁢

(

𝐱

)

+

b

⁢

g

⁢

(

𝐱

)

)

⋅

𝐡

=

a

⁢

(

D

⁢

f

)

⁢

(

𝐱

)

⋅

𝐡

+

b

⁢

(

D

⁢

g

)

⁢

(

𝐱

)

⋅

𝐡

;

“

Product

” rule

D

⁢

(

B

⁢

(

f

⁢

(

𝐱

)

,

g

⁢

(

𝐱

)

)

)

⋅

𝐡

=

B

⁢

(

(

D

⁢

f

)

⁢

(

𝐱

)

⋅

𝐡

,

g

⁢

(

𝐱

)

)

+

B

⁢

(

f

⁢

(

𝐱

)

,

(

D

⁢

g

)

⁢

(

𝐱

)

⋅

𝐡

)

,
where

B

is

bilinear

;

Chain rule

D

(

g

(

f

(

𝐱

)

)

⋅

𝐡

=

(

D

g

)

(

f

(

𝐱

)

)

⋅

(

(

D

f

)

(

𝐱

)

⋅

𝐡

)

.

Note that the derivative of

f

can be seen as a function

D

⁢

f

:

𝖵

→

L

⁢

(

𝖵

,

𝖶

)

given by

D

⁢

f

:

𝐱

↦

(

D

⁢

f

)

⁢

(

𝐱

)

, where

L

⁢

(

𝖵

,

𝖶

)

is the space of bounded linear maps from

𝖵

to

𝖶

. Since

L

⁢

(

𝖵

,

𝖶

)

can be considered a Banach space itself with the norm taken as the

operator norm

, higher derivatives can be
obtained by applying the same procedure to

D

⁢

f

and so on.

0.1

Partial derivatives

A straightforward extension of the derivatives defined above is that of

partial derivatives

for functions
of several

independent variables

. Partial derivatives have numerous applications, as for example in
physics and engineering; wave equations are among such important examples of the use of partial derivatives
in physics and engineering.

Manifolds

Let

𝖵

be a Banach space (for finite dimensional manifolds

𝖵

=

ℝ

n

). A manifold modeled on

𝖵

is a

topological space

that is

locally homeomorphic

to

𝖵

and
is endowed with enough structure to define derivatives. Since the
notion of a manifold was constructed specifically to generalize the
notion of a derivative, this seems like the end of the road for this
entry. The following discussion is rather technical, a more
intuitive explanation of the same concept can be found in the entry
on

related rates

.

Consider manifolds

V

and

W

modeled on Banach spaces

𝖵

and

𝖶

,
respectively. Say we have

y

=

f

⁢

(

x

)

for some

x

∈

V

and

y

∈

W

, then, by definition of a
manifold, we can find charts

(

X

,

𝐱

)

and

(

Y

,

𝐲

)

, where

X

and

Y

are

neighborhoods

of

x

and

y

, respectively. These charts provide us
with canonical isomorphisms between the Banach spaces

𝖵

and

𝖶

,
and the respective

tangent spaces

T

x

⁢

V

and

T

y

⁢

W

:

d

⁢

𝐱

x

:

T

x

⁢

V

→

𝖵

,

d

⁢

𝐲

y

:

T

y

⁢

W

→

𝖶

.

Now consider a map

f

:

V

→

W

between the manifolds. By
composing it with the chart maps we construct the map

g

(

X

,

𝐱

)

(

Y

,

𝐲

)

=

𝐲

∘

f

∘

𝐱

-

1

:

𝖵

→

𝖶

,

defined on an appropriately domain.
Since we now have a map between Banach spaces, we can define its
derivative at

𝐱

⁢

(

x

)

in the sense defined above, namely

D

⁢

g

(

X

,

𝐱

)

(

Y

,

𝐲

)

⁢

(

𝐱

⁢

(

x

)

)

. If this derivative exists for every
choice of

admissible

charts

(

X

,

𝐱

)

and

(

Y

,

𝐲

)

, we can say that
the derivative of

D

⁢

f

⁢

(

x

)

of

f

at

x

is defined and given by

D

⁢

f

⁢

(

x

)

=

d

⁢

𝐲

y

-

1

∘

D

⁢

g

(

X

,

𝐱

)

(

Y

,

𝐲

)

⁢

(

𝐱

⁢

(

x

)

)

∘

d

⁢

𝐱

x

(it can be shown that this is well defined and independent of the
choice of charts).

Note that the derivative is now a map between the tangent spaces of
the two manifolds

D

⁢

f

⁢

(

x

)

:

T

x

⁢

V

→

T

y

⁢

W

. Because of this a
common notation for the derivative of

f

at

x

is

T

x

⁢

f

. Another

alternative notation

for the derivative is

f

*

,

x

because of its
connection to the category-theoretical pushforward.

Distributions

Derivatives can also be generalized in less “smooth” contexts.
For example the derivative is one of
operation (

http://planetmath.org/OperationsOnDistributions

) that can be
defined for

distributions

.

Standard connection of

ℝ

n

Let

Ω

be an

open set

in

ℝ

n

. There is an operator on

vectors fields

in

Ω

which measure how a
pair of them,

X

,

Y

:

Ω

→

ℝ

n

vary, one with respect to the other:

D

X

⁢

Y

=

(

J

⁢

Y

)

⁢

X

Here

J

⁢

Y

is the Jacobian of

Y

, so when we multiply, we can see that the

components

of

D

X

⁢

Y

are the directional variations of the components of

Y

in the

direction

X

.

Additional Topic

•

Non-Newtonian calculus

Title

derivative

\metatable

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