partial derivative

partial derivative

The

partial derivative

of a multivariable function

f

is simply its

derivative

with respect to only one variable, keeping all other variables constant (which are not functions of the variable in question). The formal definition is

D

i

⁢

f

⁢

(

𝐚

)

=

∂

⁡

f

∂

⁡

a

i

=

lim

h

→

0

⁡

1

h

⁢

(

f

⁢

(

a

1

⋮

a

i

+

h

⋮

a

n

)

-

f

⁢

(

𝐚

)

)

=

lim

h

→

0

⁡

f

⁢

(

𝐚

+

h

⁢

e

→

i

)

-

f

⁢

(

𝐚

)

h

where

e

→

i

is the standard basis vector of the

i

th variable. Since this only affects the

i

th variable, one can derive the function using common rules and tables, treating all other variables (which are not functions of

a

i

) as constants. For example, if

f

⁢

(

𝐱

)

=

x

2

+

2

⁢

x

⁢

y

+

y

2

+

y

3

⁢

z

, then

(

1

)

∂

⁡

f

∂

⁡

x

=

2

⁢

x

+

2

⁢

y

(

2

)

∂

⁡

f

∂

⁡

y

=

2

⁢

x

+

2

⁢

y

+

3

⁢

y

2

⁢

z

(

3

)

∂

⁡

f

∂

⁡

z

=

y

3

Note that in equation

(

1

)

, we treated

y

as a constant, since we were differentiating with respect to

x

.

(

d

⁢

(

c

*

x

)

d

⁢

x

=

c

)

The partial derivative of a

vector-valued function

f

→

⁢

(

𝐱

)

with respect to variable

a

i

is a vector

D

i

⁢

𝐟

→

=

∂

⁡

f

→

∂

⁡

a

i

.

Multiple Partials

:

Multiple partial derivatives can be treated just like multiple derivatives. There is an additional degree of freedom though, as you can compound derivatives with respect to different variables. For example, using the above function,

(

4

)

∂

2

⁡

f

∂

⁡

x

2

=

∂

∂

⁡

x

⁢

(

2

⁢

x

+

2

⁢

y

)

=

2

(

5

)

∂

2

⁡

f

∂

⁡

z

⁢

∂

⁡

y

=

∂

∂

⁡

z

⁢

(

2

⁢

x

+

2

⁢

y

+

3

⁢

y

2

⁢

z

)

=

3

⁢

y

2

(

6

)

∂

2

⁡

f

∂

⁡

y

⁢

∂

⁡

z

=

∂

∂

⁡

y

⁢

(

y

3

)

=

3

⁢

y

2

D

12

is another way of writing

∂

∂

⁡

x

1

⁢

∂

⁡

x

2

. If

f

⁢

(

𝐱

)

is

continuous

in the

neighborhood

of

𝐱

,
and

D

i

⁢

j

⁢

f

and

D

j

⁢

i

⁢

f

are continuous in an

open set

V

, it can be shown
(see Clairaut’s theorem (

http://planetmath.org/ClairautsTheorem

)) that

D

i

⁢

j

⁢

f

⁢

(

𝐱

)

=

D

j

⁢

i

⁢

f

⁢

(

𝐱

)

in

V

, where

i

,

j

are the ith and jth variables. In fact, as long as an equal number of partials are taken with respect to each variable, changing the order of

differentiation

will produce the same results in the above condition.

Another form of notation is

f

(

a

,

b

,

c

,

…

)

⁢

(

𝐱

)

where

a

is the partial derivative with respect to the first variable

a

times,

b

is the partial with respect to the second variable

b

times, etc.

Title

partial derivative

Canonical name

PartialDerivative

Date of creation

2013-03-22 11:58:30

Last modified on

2013-03-22 11:58:30

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

26

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 26B12

Related topic

Derivative2

Related topic

DerivativeNotation

Related topic

JacobianMatrix

Related topic

DirectionalDerivative

Related topic

Gradient

Related topic

HessianMatrix

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