Editing Partial derivative (section)

{{short description|Derivative of a function with multiple variables}}
{{Calculus}}

In [[mathematics]], a '''partial derivative''' of a [[function (mathematics)#MULTIVARIATE FUNCTION|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 <math>f(x, y, \dots)</math> with respect to the variable <math>x</math> is variously denoted by
{{block indent | em = 1.2 | text = <math>f_x</math>, <math>f'_x</math>, <math>\partial_x f</math>, <math>\ D_xf</math>, <math>D_1f</math>, <math>\frac{\partial}{\partial x}f</math>, or <math>\frac{\partial f}{\partial x}</math>.}}
It can be thought of as the rate of change of the function in the <math>x</math>-direction.

Sometimes, for {{nowrap|<math>z=f(x, y, \ldots)</math>,}} the partial derivative of <math>z</math> with respect to <math>x</math> is denoted as <math>\tfrac{\partial z}{\partial x}.</math> 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:

<math display="block">f'_x(x, y, \ldots), \frac{\partial f}{\partial x} (x, y, \ldots).</math>

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,<ref name="Cajori_History_V2">{{citation |last= Cajori |first= Florian |year= 1952 |title= A History of Mathematical Notations |at= 596 |edition = 3 |volume = 2  |publisher= The Open Court Publishing Company |url=https://archive.org/details/AHistoryOfMathematicalNotationVolII/page/n153/mode/2up}}</ref> who used it for [[partial difference equation|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.<ref name="miller_earliest">{{citation |last= Miller |first= Jeff |date= n.d. |contribution= Earliest Uses of Symbols of Calculus |contribution-url= https://mathshistory.st-andrews.ac.uk/Miller/mathsym/calculus/ |access-date= 2023-06-15 |editor-last= O'Connor |editor-first= John J. |editor2-last= Robertson |editor2-first= Edmund F. |editor2-link= Edmund F. Robertson |title= [[MacTutor History of Mathematics archive]] |publisher= [[University of St Andrews]] |mode= cs1}}</ref>