Partial derivatives

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.
The partial derivative of a function



f
(
x
,
y
,
…
)


{\displaystyle f(x,y,\dots )}
with respect to the variable



x


{\displaystyle x}
is variously denoted by





f

x

′

,

f

x


,

∂

x


f
,


D

x


f
,

D

1


f
,


∂

∂
x



f
,

or




∂
f


∂
x



.


{\displaystyle f'_{x},f_{x},\partial _{x}f,\ D_{x}f,D_{1}f,{\frac {\partial }{\partial x}}f,{\text{ or }}{\frac {\partial f}{\partial x}}.}
Sometimes, for



z
=
f
(
x
,
y
,
…
)
,


{\displaystyle z=f(x,y,\ldots ),}
the partial derivative of



z


{\displaystyle z}
with respect to



x


{\displaystyle x}
is denoted as







∂
z


∂
x




.


{\displaystyle {\tfrac {\partial z}{\partial x}}.}
Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in:





f

x


(
x
,
y
,
…
)
,



∂
f


∂
x



(
x
,
y
,
…
)
.


{\displaystyle f_{x}(x,y,\ldots ),{\frac {\partial f}{\partial x}}(x,y,\ldots ).}
The symbol used to denote partial derivatives is ∂. One of the first known uses of this symbol in mathematics is by Marquis de Condorcet from 1770, who used it for partial differences. The modern partial derivative notation was created by Adrien-Marie Legendre (1786) (although he later abandoned it, Carl Gustav Jacob Jacobi reintroduced the symbol in 1841).

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