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{{short description|Notation of differential calculus}}
{{Lead too short|date=March 2023}}
{{Calculus |Differential}}

In [[differential calculus]], there is no single uniform '''notation for differentiation'''. Instead, various notations for the [[derivative]] of a [[function (mathematics)|function]] or [[dependent variable|variable]] have been proposed by various mathematicians. The usefulness of each notation varies with the context, and it is sometimes advantageous to use more than one notation in a given context. The most common notations for differentiation (and its opposite operation, the [[antiderivative|antidifferentiation]] or [[antiderivative|indefinite integration]]) are listed below.

== Leibniz's notation ==
{{main|Leibniz's notation}}

The original notation employed by [[Gottfried Leibniz]] is used throughout mathematics. It is particularly common when the equation {{math|''y'' {{=}} ''f''(''x'')}} is regarded as a functional relationship between [[dependent and independent variables]] {{math|''y''}} and {{math|''x''}}. Leibniz's notation makes this relationship explicit by writing the derivative as:{{r|vpr}}
<math display="block">\frac{dy}{dx}.</math>
Furthermore, the derivative of {{math|''f''}} at {{math|''x''}} is therefore written
<math display="block">\frac{df}{dx}(x)\text{ or }\frac{d f(x)}{dx}\text{ or }\frac{d}{dx} f(x).</math>

Higher derivatives are written as:{{sfnp|Varberg|Purcell|Rigdon|2007|p=125&ndash;126}}
<math display="block">\frac{d^2y}{dx^2}, \frac{d^3y}{dx^3}, \frac{d^4y}{dx^4}, \ldots, \frac{d^ny}{dx^n}.</math>
This is a suggestive notational device that comes from formal manipulations of symbols, as in,
<math display="block">\frac{d\left(\frac{dy}{dx}\right)}{dx} = \left(\frac{d}{dx}\right)^2y = \frac{d^2y}{dx^2}.</math>

The value of the derivative of {{math|''y''}} at a point {{math|''x'' {{=}} ''a''}} may be expressed in two ways using Leibniz's notation:
<math display="block">\left.\frac{dy}{dx}\right|_{x=a} \text{ or } \frac{dy}{dx}(a).</math>

Leibniz's notation allows one to specify the variable for differentiation (in the denominator). This is especially helpful when considering [[partial derivative]]s. It also makes the [[chain rule]] easy to remember and recognize:
<math display="block">\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}.</math>

Leibniz's notation for differentiation does not require assigning meaning to symbols such as {{math|''dx''}} or {{math|''dy''}} (known as [[Differential_(mathematics)|differentials]]) on their own, and some authors do not attempt to assign these symbols meaning.{{r|vpr}}  Leibniz treated these symbols as [[infinitesimal]]s.  Later authors have assigned them other meanings, such as infinitesimals in [[non-standard analysis]], or [[exterior derivative]]s. Commonly, {{math|''dx''}} is left undefined or equated with <math>\Delta x</math>, while  {{math|''dy''}} is assigned a meaning in terms of {{math|''dx''}}, via the equation

: <math>dy = \frac{dy}{dx} \cdot dx, </math>

which may also be written, e.g.

: <math>df = f'(x) \cdot dx </math>

(see [[#Lagrange's notation|below]]). Such equations give rise to the terminology found in some texts wherein the derivative is referred to as the "differential coefficient" (i.e., the [[coefficient]] of {{math|''dx''}}).

Some authors and journals set the differential symbol {{math|''d''}} in [[roman type]] instead of [[italic type|italic]]: {{math|d''x''}}. The [[ISO/IEC 80000]] scientific style guide recommends this style.

== Lagrange's notation ==
{{image frame|width=200|innerstyle=font-size:400%; line-height: 120%; font-family:Times New Roman, serif; text-align:center;|
caption = A function <var>f</var> of <var>x</var>, differentiated once in Lagrange's notation. |
content=''f''{{′}}(''x'')}}

One of the most common modern notations for differentiation is named after [[Joseph Louis Lagrange]], even though it was actually invented by [[Leonhard Euler|Euler]] and just popularized by the former.  In Lagrange's notation, a [[prime (symbol)|prime mark]] denotes a derivative.  If ''f'' is a function, then its derivative evaluated at ''x'' is written
:<math>f'(x)</math>.
It first appeared in print in 1749.<ref>{{Cite journal|url=https://babel.hathitrust.org/cgi/pt?id=ucm.532509502x&view=1up&seq=520|journal =[[Acta Eruditorum|Nova Acta Eruditorum]] |page=512|date=September 1749|title=Notation for differentiation|last1 = Grosse|first1 = Johann|last2 = Breitkopf|first2 = Bernhard Christoph|last3 = Martin|first3 = Johann Christian|last4 = Gleditsch|first4 = Johann Friedrich}}</ref>

Higher derivatives are indicated using additional prime marks, as in <math>f''(x)</math> for the [[second derivative]] and <math>f'''(x)</math> for the [[third derivative]].  The use of repeated prime marks eventually becomes unwieldy.  Some authors continue by employing [[Roman numeral]]s, usually in lower case,<ref>{{Cite book|title=Fundamentals of calculus|last=Morris, Carla C.|others=Stark, Robert M., 1930-2017.|isbn=9781119015314|location=Hoboken, New Jersey|oclc=893974565|date = 2015-07-28}}</ref><ref>{{Cite book|url=https://archive.org/details/differentialand00osbogoog|title=Differential and Integral Calculus|last=Osborne|first=George A.|publisher=D. C. Heath and co.|year=1908|location=Boston|pages=[https://archive.org/details/differentialand00osbogoog/page/n78 63]-65}}</ref> as in
:<math>f^{\mathrm{iv}}(x), f^{\mathrm{v}}(x), f^{\mathrm{vi}}(x), \ldots,</math>
to denote fourth, fifth, sixth, and higher order derivatives.  Other authors use Arabic numerals in parentheses, as in
:<math>f^{(4)}(x), f^{(5)}(x), f^{(6)}(x), \ldots.</math>
This notation also makes it possible to describe the ''n''th derivative, where ''n'' is a variable.  This is written
:<math>f^{(n)}(x).</math>

Unicode characters related to Lagrange's notation include
* {{unichar|2032|PRIME|cwith=◌|note=derivative}}
* {{unichar|2033|DOUBLE PRIME|cwith=◌|note=double derivative}}
* {{unichar|2034|TRIPLE PRIME|cwith=◌|note=third derivative}}
* {{unichar|2057|QUADRUPLE PRIME|cwith=◌|note=fourth derivative}}

When there are two independent variables for a function ''f''(''x'',&thinsp;''y''), the following convention may be followed:<ref name="DeMorgan">''The Differential and Integral Calculus'' ([[Augustus De Morgan]], 1842). pp. 267-268</ref><!-- appears as z(x,y) for a function φ(x,y,z) = 0 -->
: <math>\begin{align}
          f^\prime &= \frac{\partial f}{\partial x} = f_x \\[5pt]
          f_\prime &= \frac{\partial f}{\partial y} = f_y \\[5pt]
  f^{\prime\prime} &= \frac{\partial ^2 f}{\partial x^2} = f_{xx} \\[5pt]
   f_\prime^\prime &= \frac{\partial ^2 f}{\partial y \partial x}\ = f_{xy} \\[5pt]
  f_{\prime\prime} &= \frac{\partial ^2 f}{\partial y^2} = f_{yy}
\end{align}</math>

=== Lagrange's notation for antidifferentiation ===
{{image frame|width=200|innerstyle=font-size:300%; line-height: 120%; font-family:Times New Roman, serif; text-align:center;|
caption = The single and double indefinite integrals of <var>f</var> with respect to <var>x</var>, in the Lagrange notation. |
content = ''f''<sup style{{=}}"padding-left:0.3em;">(&minus;1)</sup>(''x'')<br/>''f''<sup style{{=}}"padding-left:0.3em;">(&minus;2)</sup>(''x'')}}

When taking the antiderivative, Lagrange followed Leibniz's notation:<ref name="Lagrange">[[Joseph Louis Lagrange|Lagrange]], ''Nouvelle méthode pour résoudre les équations littérales par le moyen des séries'' (1770), p. 25-26. http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN308900308|LOG_0017&physid=PHYS_0031</ref>
:<math>f(x) = \int f'(x)\,dx = \int y'\,dx.</math>

However, because integration is the inverse operation of differentiation, Lagrange's notation for higher order derivatives extends to integrals as well.  Repeated integrals of ''f'' may be written as
:<math>f^{(-1)}(x)</math> for the first integral (this is easily confused with the [[inverse function]] <math>f^{-1}(x)</math>),
:<math>f^{(-2)}(x)</math> for the second integral,
:<math>f^{(-3)}(x)</math> for the third integral, and
:<math>f^{(-n)}(x)</math> for the ''n''th integral.

== D-notation ==
{{image frame|width=200|innerstyle=font-size:400%; line-height: 120%; font-family:Times New Roman, serif; text-align:center;|
caption = The <var>x</var> derivative of <var>y</var> and the second derivative of <var>f</var>, Euler notation. |
content = ''D{{sub|x}}y''<br/>''D''{{i sup|2}}''f''}}

This notation is sometimes called '''{{vanchor|Euler's notation}}''' although it was introduced by [[Louis François Antoine Arbogast]],<ref>
    {{cite web
    |last        = Cajori
    |first       = Florian
    |date        = 1923
    |title       = The History of Notations of the Calculus
    |publisher   = Mathematics Department, Princeton University
    |url         = https://www.jstor.org/stable/1967725
    |jstor       = 1967725
    |page        = 7
    |access-date = 2025-01-07
    }}
</ref> and it seems that [[Leonhard Euler]] did not use it.{{citation needed|date=May 2024}}

This notation uses a [[differential operator]] denoted as {{math|''D''}} ('''D operator''')<ref>{{cite web|url=http://www.codecogs.com/library/maths/calculus/differential/the-d-operator.php|title=The D operator - Differential - Calculus - Maths Reference with Worked Examples|website=www.codecogs.com|url-status=live|archive-url=https://web.archive.org/web/20160119050319/http://www.codecogs.com/library/maths/calculus/differential/the-d-operator.php|archive-date=2016-01-19}}</ref>{{Not in citation|date=March 2023}} or {{math|''D̃''}} ('''Newton–Leibniz operator''').<ref name="EulerMathWorld">Weisstein, Eric W. "Differential Operator." From ''MathWorld''--A Wolfram Web Resource. {{cite web |url=http://mathworld.wolfram.com/DifferentialOperator.html |title=Differential Operator |access-date=2016-02-07 |url-status=live |archive-url=https://web.archive.org/web/20160121215815/http://mathworld.wolfram.com/DifferentialOperator.html |archive-date=2016-01-21 }}</ref>  When applied to a function {{math|''f''(''x'')}}, it is defined by
:<math>(Df)(x) = \frac{df(x)}{dx}.</math>
Higher derivatives are notated as "powers" of ''D'' (where the superscripts denote iterated [[function composition|composition]] of ''D''), as in<ref name="DeMorgan" />
:<math>D^2f</math> for the second derivative,
:<math>D^3f</math> for the third derivative, and
:<math>D^nf</math> for the ''n''th derivative.

D-notation leaves implicit the variable with respect to which differentiation is being done. However, this variable can also be made explicit by putting its name as a subscript: if ''f'' is a function of a variable ''x'', this is done by writing<ref name="DeMorgan"/>
:<math>D_x f</math> for the first derivative,
:<math>D^2_x f</math> for the second derivative,
:<math>D^3_x f</math> for the third derivative, and
:<math>D^n_x f</math> for the ''n''th derivative.
When ''f'' is a function of several variables, it is common to use "[[∂]]", a stylized cursive lower-case d, rather than "{{math|''D''}}". As above, the subscripts denote the derivatives that are being taken. For example, the second partial derivatives of a function {{math|''f''(''x'', ''y'')}} are:<ref name="DeMorgan"/>
:<math>
\begin{align}
& \partial_{xx} f = \frac{\partial^2 f}{\partial x^2}, \\[5pt]
& \partial_{xy} f = \frac{\partial^2 f}{\partial y\,\partial x}, \\[5pt]
& \partial_{yx} f = \frac{\partial^2 f}{\partial x\,\partial y}, \\[5pt]
& \partial_{yy} f = \frac{\partial^2 f}{\partial y^2}.
\end{align}
</math>
See {{section link||Partial derivatives}}.

D-notation is useful in the study of  [[differential equation]]s and in [[differential algebra]].

=== D-notation for antiderivatives ===
{{image frame|width=200|innerstyle=font-size:400%; line-height: 120%; font-family:Times New Roman, serif; text-align:center;|
caption = The <var>x</var> antiderivative of <var>y</var> and the second antiderivative of <var>f</var>, Euler notation. |
content = ''D''{{su|p=&minus;1|b=''x''}}''y''<br/>''D''{{i sup|&minus;2}}''f''}}

D-notation can be used for antiderivatives in the same way that Lagrange's notation is<ref>Weisstein, Eric W. "Repeated Integral." From ''MathWorld''--A Wolfram Web Resource. {{cite web |url=http://mathworld.wolfram.com/RepeatedIntegral.html |title=Repeated Integral |access-date=2016-02-07 |url-status=live |archive-url=https://web.archive.org/web/20160201051403/http://mathworld.wolfram.com/RepeatedIntegral.html |archive-date=2016-02-01 }}</ref> as follows<ref name="EulerMathWorld" />
:<math>D^{-1}f(x)</math> for a first antiderivative,
:<math>D^{-2}f(x)</math> for a second antiderivative, and
:<math>D^{-n}f(x)</math> for an ''n''th antiderivative.

==Newton's notation{{anchor|Newton's notation}}==
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{{image frame|width=200|innerstyle=font-size:400%; line-height: 120%; font-family:Times New Roman, serif; display:flex;justify-content:space-evenly;align-items:center;|
caption = The first and second derivatives of <var>x</var>, Newton's notation.|
content = <span>''ẋ''</span><span>''ẍ''</span>}}

[[Isaac Newton]]'s notation for differentiation (also called the '''dot notation''', '''[[fluxion|fluxions]]''', or sometimes, crudely, the '''flyspeck notation'''<ref>{{cite book |chapter-url=https://books.google.com/books?id=BnArjLNjXuYC&q=Flyspeck+notation&pg=PA3 |title=A First Course in Differential Equations |first=Dennis G. |last=Zill |chapter=1.1 |page=3 |edition=9th |publisher=[[Brooks/Cole]] |location=[[Belmont, CA]] |year=2009 |isbn=978-0-495-10824-5}}</ref> for differentiation) places a dot over the dependent variable.  That is, if ''y'' is a function of ''t'', then the derivative of ''y'' with respect to ''t'' is
:<math>\dot y</math>
Higher derivatives are represented using multiple dots, as in
:<math>\ddot y, \overset{...}{y}</math>
Newton extended this idea quite far:<ref>Newton's notation reproduced from:

*1st to 5th derivatives: ''Quadratura curvarum'' ([[Isaac Newton|Newton]], 1704), p. 7 (p. 5r in original MS: {{cite web|title=Newton Papers : On the Quadrature of Curves|url=http://cudl.lib.cam.ac.uk/view/MS-ADD-03962/9|url-status=live|archive-url=https://web.archive.org/web/20160228002251/http://cudl.lib.cam.ac.uk/view/MS-ADD-03962/9|archive-date=2016-02-28|access-date=2016-02-05}}).
*1st to 7th, ''n''th and (''n''+1)th derivatives: ''Method of Fluxions'' ([[Isaac Newton|Newton]], 1736), pp. 313-318 and p. 265 (p. 163 in original MS: {{cite web|title=Newton Papers : Fluxions|url=http://cudl.lib.cam.ac.uk/view/MS-ADD-03960/257|url-status=live|archive-url=https://web.archive.org/web/20170406183849/http://cudl.lib.cam.ac.uk/view/MS-ADD-03960/257|archive-date=2017-04-06|access-date=2016-02-05}})
*1st to 5th derivatives : ''A Treatise of Fluxions'' (Colin MacLaurin, 1742), p. 613
*1st to 4th and ''n''th derivatives: Articles "Differential" and "Fluxion", ''Dictionary of Pure and Mixed Mathematics'' (Peter Barlow, 1814)
*1st to 4th, 10th and ''n''th derivatives: Articles 622, 580 and 579 in ''[[A History of Mathematical Notations]]'' (F .Cajori, 1929)
*1st to 6th and ''n''th derivatives: ''The Mathematical Papers of Isaac Newton'' Vol. 7 1691-1695 (D. T. Whiteside, 1976), pp.88 and 17
*1st to 3rd and ''n''th derivatives: ''A History of Analysis'' (Hans Niels Jahnke, 2000), pp. 84-85
The dot for ''n''th derivative may be omitted ( <math>\overset{\,n}{y}</math> )
</ref>
:<math>\begin{align}
                 \ddot{y} &\equiv \frac{d^2y}{dt^2} = \frac{d}{dt}\left(\frac{dy}{dt}\right)  = \frac{d}{dt}\Bigl(\dot{y}\Bigr) = \frac{d}{dt}\Bigl(f'(t)\Bigr) = D_t^2 y = f''(t) = y''_t \\[5pt]
         \overset{...}{y} &= \dot{\ddot{y}} \equiv \frac{d^3y}{dt^3} = D_t^3 y = f'''(t) = y'''_t \\[5pt]
   \overset{\,4}{\dot{y}} &= \overset{....}{y} = \ddot{\ddot{y}} \equiv \frac{d^4y}{dt^4} = D_t^4 y = f^{\rm IV}(t) = y^{(4)}_t \\[5pt]
   \overset{\,5}{\dot{y}} &= \ddot{\overset{...}{y}} = \dot{\ddot{\ddot{y}}} = \ddot{\dot{\ddot{y}}} \equiv \frac{d^5y}{dt^5} = D_t^5 y = f^{\rm V}(t) = y^{(5)}_t \\[5pt]
   \overset{\,6}{\dot{y}} &= \overset{...}{\overset{...}{y}} \equiv \frac{d^6y}{dt^6} = D_t^6 y = f^{\rm VI}(t) = y^{(6)}_t \\[5pt]
   \overset{\,7}{\dot{y}} &= \dot{\overset{...}{\overset{...}{y}}} \equiv \frac{d^7y}{dt^7} = D_t^7 y = f^{\rm VII}(t) = y^{(7)}_t \\[5pt]
  \overset{\,10}{\dot{y}} &= \ddot{\ddot{\ddot{\ddot{\ddot{y}}}}} \equiv \frac{d^{10}y}{dt^{10}} = D_t^{10} y = f^{\rm X}(t) = y^{(10)}_t \\[5pt]
   \overset{\,n}{\dot{y}} &\equiv \frac{d^ny}{dt^n} = D_t^n y = f^{(n)}(t) = y^{(n)}_t
\end{align}</math>

Unicode characters related to Newton's notation include:
* {{unichar|0307|COMBINING DOT ABOVE|cwith=◌|note=derivative}}
* {{unichar|0308|COMBINING DIAERESIS|cwith=◌|note=double derivative}}
* {{unichar|20DB|COMBINING THREE DOTS ABOVE|cwith=◌|note=third derivative}} ← replaced by "combining diaeresis" + "combining dot above".
* {{unichar|20DC|COMBINING FOUR DOTS ABOVE|cwith=◌|note=fourth derivative}} ← replaced by "combining diaeresis" twice.
* {{unichar|030D|COMBINING VERTICAL LINE ABOVE|cwith=◌|note=integral}}
* {{unichar|030E|COMBINING DOUBLE VERTICAL LINE ABOVE|cwith=◌|note=second integral}}
* {{unichar|25AD|WHITE RECTANGLE|note=integral}}
* {{unichar|20DE|COMBINING ENCLOSING SQUARE|cwith=◌|note=integral}}
* {{unichar|1DE0|COMBINING LATIN SMALL LETTER N|cwith=◌|note=''n''th derivative}}

Newton's notation is generally used when the independent variable denotes [[time]].  If location {{math|''y''}} is a function of ''t'', then <math>\dot y</math> denotes [[velocity]]<ref>Weisstein, Eric W. "Overdot." From ''MathWorld''--A Wolfram Web Resource. {{cite web |url=http://mathworld.wolfram.com/Overdot.html |title=Overdot |access-date=2016-02-05 |url-status=live |archive-url=https://web.archive.org/web/20150905171914/http://mathworld.wolfram.com/Overdot.html |archive-date=2015-09-05 }}</ref> and <math>\ddot y</math> denotes [[acceleration]].<ref>Weisstein, Eric W. "Double Dot." From ''MathWorld''--A Wolfram Web Resource. {{cite web |url=http://mathworld.wolfram.com/DoubleDot.html |title=Double Dot |access-date=2016-02-05 |url-status=live |archive-url=https://web.archive.org/web/20160303174226/http://mathworld.wolfram.com/DoubleDot.html |archive-date=2016-03-03 }}</ref> This notation is popular in [[physics]] and [[mathematical physics]].  It also appears in areas of mathematics connected with physics such as [[differential equation]]s.

When taking the derivative of a dependent variable ''y'' = ''f''(''x''), an alternative notation exists:<ref>Article 580 in Florian Cajori, ''A History of Mathematical Notations'' (1929), Dover Publications, Inc. New York. {{isbn|0-486-67766-4}}</ref>
:<math>\frac{\dot{y}}{\dot{x}} = \dot{y}:\dot{x} \equiv \frac{dy}{dt}:\frac{dx}{dt} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{dy}{dx} = \frac{d}{dx}\Bigl(f(x)\Bigr) = D y = f'(x) = y'.</math>

Newton developed the following partial differential operators using side-dots on a curved X ( ⵋ ). Definitions given by Whiteside are below:<ref>"Patterns of Mathematical Thought in the Later Seventeenth Century", ''Archive for History of Exact Sciences'' Vol. 1, No. 3 (D. T. Whiteside, 1961), pp. 361-362,378</ref><ref>S.B. Engelsman has given more strict definitions in ''Families of Curves and the Origins of Partial Differentiation'' (2000), pp. 223-226</ref>

: <!-- shape of X must be: ⵋ --><math>\begin{align}
       \mathcal{X}                                                            \ &=\  f(x,y) \,, \\[5pt]
       \cdot\mathcal{X}                                                       \ &=\  x\frac{\partial f}{\partial x} = xf_x\,, \\[5pt]
       \mathcal{X}\!\cdot                                                     \ &=\  y\frac{\partial f}{\partial y} = yf_y\,, \\[5pt]
       \colon\!\mathcal{X}\,\text{ or }\,\cdot\!\left(\cdot\mathcal{X}\right) \ &=\  x^2\frac{\partial^2 f}{\partial x^2} = x^2 f_{xx}\,, \\[5pt]
       \mathcal{X}\colon\,\text{ or }\,\left(\mathcal{X}\cdot\right)\!\cdot   \ &=\  y^2\frac{\partial^2 f}{\partial y^2} = y^2 f_{yy}\,, \\[5pt]
       \cdot\mathcal{X}\!\cdot\                                               \ &=\  xy\frac{\partial^2 f}{\partial x \, \partial y} = xy f_{xy}\,,
\end{align}</math>

=== Newton's notation for integration ===
{{image frame|width=200|innerstyle=font-size:400%; line-height: 120%; font-family:Times New Roman, serif; display:flex;justify-content:space-evenly;align-items:center;|
caption = The first and second antiderivatives of <var>x</var>, in one of Newton's notations.|
content = <span>''x̍''</span><span>''x̎''</span>}}

Newton developed many different notations for [[integral|integration]] in his ''Quadratura curvarum'' (1704) and [[Method of Fluxions|later works]]: he wrote a small vertical bar or prime above the dependent variable ({{math|''y̍''}} ), a prefixing rectangle ({{math|▭''y''}}), or the inclosure of the term in a rectangle (<span style="border-style: solid; border-width: 1.5px 1.5px 1.5px 1.5px; padding-left: 4px; padding-right: 4px;">{{math|''y''}}</span>) to denote the ''[[Method of Fluxions|fluent]]'' or time integral ([[absement]]).

: <math>\begin{align}
                      y &= \Box \dot{y} \equiv \int \dot{y} \,dt = \int f'(t) \,dt = D_t^{-1} (D_t y) = f(t) + C_0 = y_t + C_0 \\
  \overset{\,\prime}{y} &= \Box y \equiv \int y \,dt = \int f(t) \,dt = D_t^{-1} y = F(t) + C_1
\end{align}</math>

To denote multiple integrals, Newton used two small vertical bars or primes ({{math|''y̎''}}), or a combination of previous symbols {{math|▭''y̍''}}&thinsp;<span style="border-style: solid; border-width: 1.5px 1.5px 1.5px 1.5px; padding-left: 4px; padding-right: 4px;">{{math|''y̍''}}</span>, to denote the second time integral (absity).

: <math>\overset{\,\prime\prime}{y} = \Box \overset{\,\prime}{y} \equiv \int \overset{\,\prime}{y} \,dt = \int F(t) \,dt = D_t^{-2} y = g(t) + C_2</math>

Higher order time integrals were as follows:<ref>Newton's notation for integration reproduced from:
* 1st to 3rd integrals: ''Quadratura curvarum'' ([[Isaac Newton|Newton]], 1704), p. 7 (p. 5r in original MS: {{cite web |url=http://cudl.lib.cam.ac.uk/view/MS-ADD-03962/9 |title=Newton Papers : On the Quadrature of Curves |access-date=2016-02-05 |url-status=live |archive-url=https://web.archive.org/web/20160228002251/http://cudl.lib.cam.ac.uk/view/MS-ADD-03962/9 |archive-date=2016-02-28 }})
* 1st to 3rd integrals: ''Method of Fluxions'' ([[Isaac Newton|Newton]], 1736), pp. 265-266 (p. 163 in original MS: {{cite web |url=http://cudl.lib.cam.ac.uk/view/MS-ADD-03960/257 |title=Newton Papers : Fluxions |access-date=2016-02-05 |url-status=live |archive-url=https://web.archive.org/web/20170406183849/http://cudl.lib.cam.ac.uk/view/MS-ADD-03960/257 |archive-date=2017-04-06 }})
* 4th integrals: ''The Doctrine of Fluxions'' (James Hodgson, 1736), pp. 54 and 72
* 1st to 2nd integrals: Articles 622 and 365 in ''A History of Mathematical Notations'' (F .Cajori, 1929)
The ''n''th integral notation is deducted from the ''n''th derivative. It could be used in ''Methodus Incrementorum Directa & Inversa'' (Brook Taylor, 1715)</ref>
: <math>\begin{align}
        \overset{\,\prime\prime\prime}{y} &= \Box \overset{\,\prime\prime}{y} \equiv \int \overset{\,\prime\prime}{y} \,dt = \int g(t) \,dt = D_t^{-3} y = G(t) + C_3 \\
  \overset{\,\prime\prime\prime\prime}{y} &= \Box \overset{\,\prime\prime\prime}{y} \equiv \int \overset{\,\prime\prime\prime}{y} \,dt = \int G(t) \,dt = D_t^{-4} y = h(t) + C_4 \\
       \overset{\;n}\overset{\,\prime}{y} &= \Box \overset{\;n-1}\overset{\,\prime}y \equiv \int \overset{\;n-1}\overset{\,\prime}y \,dt = \int s(t) \,dt = D_t^{-n} y = S(t) + C_n
\end{align}</math>

This [[mathematical notation]] did not become widespread because of printing difficulties<sup>[''Citation needed'']</sup> and the [[Leibniz–Newton calculus controversy]].

== Partial derivatives ==
{{image frame|width=200|innerstyle=font-size:400%; line-height: 120%; font-family:Times New Roman, serif; display:flex;justify-content:space-evenly;align-items:center;|
caption = A function <var>f</var> differentiated against <var>x</var>, then against <var>x</var> and <var>y</var>.|
content = <span>''f{{sub|x}}''</span><span>''f{{sub|xy}}''</span>}}

When more specific types of differentiation are necessary, such as in [[multivariate calculus]] or [[tensor analysis]], other notations are common.

For a function ''f'' of a single independent variable ''x'', we can express the derivative using subscripts of the independent variable:

: <math>\begin{align}
      f_x &= \frac{df}{dx} \\[5pt]
  f_{x x} &= \frac{d^2f}{dx^2}.
\end{align}</math>

This type of notation is especially useful for taking [[partial derivative]]s of a function of several variables.

{{image frame|width=200|innerstyle=font-size:400%; line-height: 120%; font-family:Times New Roman, serif; text-align:center;|
content = {{sfrac|''∂f''|''∂x''}} |
caption = A function <var>f</var> differentiated against <var>x</var>.}}

Partial derivatives are generally distinguished from ordinary derivatives by replacing the differential operator ''d'' with a "[[∂]]" symbol. For example, we can indicate the partial derivative of {{nowrap|''f''(''x'',&thinsp;''y'',&thinsp;''z'')}} with respect to ''x'', but not to ''y'' or ''z'' in several ways:
:<math>\frac{\partial f}{\partial x} = f_x = \partial_x f.</math>
What makes this distinction important is that a non-partial derivative such as <math>\textstyle \frac{df}{dx}</math> ''may'', depending on the context, be interpreted as a rate of change in <math>f</math> relative to <math>x</math> when all variables are allowed to vary simultaneously, whereas with a partial derivative such as <math>\textstyle \frac{\partial f}{\partial x}</math> it is explicit that only one variable should vary.

Other notations can be found in various subfields of mathematics, physics, and engineering; see for example the [[Maxwell relations]] of [[thermodynamics]]. The symbol <math>\left(\frac{\partial T}{\partial V}\right)_{\!S} </math> is the derivative of the temperature ''T'' with respect to the volume ''V'' while keeping constant the entropy (subscript) ''S'', while <math>\left(\frac{\partial T}{\partial V}\right)_{\!P} </math> is the derivative of the temperature with respect to the volume while keeping constant the pressure ''P''. This becomes necessary in situations where the number of variables exceeds the degrees of freedom, so that one has to choose which other variables are to be kept fixed.

Higher-order partial derivatives with respect to one variable are expressed as
:<math>
\begin{align}
& \frac{\partial^2f}{\partial x^2} = f_{xx}, \\[5pt]
& \frac{\partial^3f}{\partial x^3} = f_{xxx},
\end{align}
</math>
and so on. Mixed partial derivatives can be expressed as

:<math>\frac{\partial^2 f}{\partial y  \, \partial x} = f_{xy}.</math>

In this last case the variables are written in inverse order between the two notations, explained as follows:

:<math>
\begin{align}
& (f_x)_y = f_{xy}, \\[5pt]
& \frac{\partial}{\partial y}\!\left(\frac{\partial f}{\partial x}\right) = \frac{\partial^2f}{\partial y \, \partial x}.
\end{align}
</math>
So-called [[multi-index notation]] is used in situations when the above notation becomes cumbersome or insufficiently expressive. When considering functions on <math>\R^n</math>, we define a multi-index to be an ordered list of <math>n</math> non-negative integers: <math>\alpha = (\alpha_1,\ldots,\alpha_n), \ \alpha_i \in \Z_{\geq 0}</math>. We then define, for <math>f:\R^n \to X</math>, the notation

: <math>\partial^\alpha f = \frac{\partial^{\alpha_1}}{\partial x_1^{\alpha_1}} \cdots \frac{\partial^{\alpha_n}}{\partial x_n^{\alpha_n}} f</math>

In this way some results (such as the [[Leibniz rule (generalized product rule)|Leibniz rule]]) that are tedious to write in other ways can be expressed succinctly -- some examples can be found in the [[Multi-index notation|article on multi-indices]].<ref>{{Cite book|last=Tu|first=Loring W.|url=https://www.worldcat.org/oclc/682907530|title=An introduction to manifolds|date=2011|publisher=Springer|isbn=978-1-4419-7400-6|edition=2|location=New York|oclc=682907530}}</ref>

== Notation in vector calculus ==

[[Vector calculus]] concerns [[derivative|differentiation]] and [[integral|integration]] of [[vector field|vector]] or [[scalar fields]].  Several notations specific to the case of three-dimensional [[Euclidean space]] are common.

Assume that {{math|(''x'', ''y'', ''z'')}} is a given [[Cartesian coordinate system]], that '''A''' is a [[vector field]] with components <math>\mathbf{A} = (\mathbf{A}_x, \mathbf{A}_y, \mathbf{A}_z)</math>, and that <math>\varphi = \varphi(x,y,z)</math> is a [[scalar field]].

The differential operator introduced by [[William Rowan Hamilton]], written [[nabla symbol|∇]] and called [[del]] or nabla, is symbolically defined in the form of a vector,
:<math>\nabla = \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right)\!,</math>
where the terminology ''symbolically'' reflects that the operator ∇ will also be treated as an ordinary vector.

{{image frame|width=200|innerstyle=font-size:400%; line-height: 120%; font-family:Times New Roman, serif; text-align:center;|
content = ∇''φ'' | caption=Gradient of the scalar field ''φ''. }}
* '''[[Gradient]]''': The gradient <math>\mathrm{grad\,} \varphi</math> of the scalar field <math>\varphi</math> is a vector, which is symbolically expressed by the [[scalar multiplication|multiplication]] of ∇ and scalar field ''<math>\varphi</math>'',

::<math>\begin{align}
  \operatorname{grad} \varphi
    &= \left( \frac{\partial \varphi}{\partial x}, \frac{\partial \varphi}{\partial y}, \frac{\partial \varphi}{\partial z} \right) \\
    &= \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right) \varphi \\
    &= \nabla \varphi
\end{align}</math>

{{image frame|width=200|innerstyle=font-size:400%; line-height: 120%; font-family:Times New Roman, serif; text-align:center;|
content = ∇∙'''A''' | caption=The divergence of the vector field  '''A'''.}}
* '''[[Divergence]]''': The divergence <math>\mathrm{div}\,\mathbf{A}</math> of the vector field '''A''' is a scalar, which is symbolically expressed by the [[dot product]] of ∇ and the vector '''A''',

:: <math>\begin{align}
  \operatorname{div} \mathbf{A}
    &= {\partial A_x \over \partial x} + {\partial A_y \over \partial y} + {\partial A_z \over \partial z} \\
    &= \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right)  \cdot \mathbf{A} \\
    &= \nabla \cdot \mathbf{A}
\end{align}</math>

{{image frame|width=200|innerstyle=font-size:400%; line-height: 120%; font-family:Times New Roman, serif; text-align:center;|
content = ∇{{sup|2}}''φ'' | caption = The Laplacian of the scalar field ''φ''.}}
* '''[[Laplacian]]''': The Laplacian <math>\operatorname{div} \operatorname{grad} \varphi</math> of the scalar field <math>\varphi</math> is a scalar, which is symbolically expressed by the scalar multiplication of ∇<sup>2</sup> and the scalar field ''φ'',

:: <math>\begin{align}
  \operatorname{div} \operatorname{grad} \varphi
    &= \nabla \cdot (\nabla \varphi) \\
    &= (\nabla \cdot \nabla) \varphi \\
    &= \nabla^2 \varphi \\
    &= \Delta \varphi \\
\end{align}</math>

{{image frame|width=200|innerstyle=font-size:400%; line-height: 120%; font-family:Times New Roman, serif; text-align:center;|
content = ∇×'''A''' | caption= The curl of vector field '''A'''.}}
* '''[[Curl (mathematics)|Rotation]]''': The rotation <math>\mathrm{curl}\,\mathbf{A}</math>, or <math>\mathrm{rot}\,\mathbf{A}</math>, of the vector field '''A''' is a vector, which is symbolically expressed by the [[cross product]] of ∇ and the vector '''A''',

:: <math>\begin{align}
  \operatorname{curl} \mathbf{A}
    &= \left(
         {\partial A_z \over {\partial y} } - {\partial A_y \over {\partial z} },
         {\partial A_x \over {\partial z} } - {\partial A_z \over {\partial x} },
         {\partial A_y \over {\partial x} } - {\partial A_x \over {\partial y} }
       \right) \\
    &= \left( {\partial A_z \over {\partial y} } - {\partial A_y \over {\partial z} } \right) \mathbf{i} +
       \left( {\partial A_x \over {\partial z} } - {\partial A_z \over {\partial x} } \right) \mathbf{j} +
       \left( {\partial A_y \over {\partial x} } - {\partial A_x \over {\partial y} } \right) \mathbf{k} \\
    &= \begin{vmatrix}
         \mathbf{i} & \mathbf{j} & \mathbf{k} \\
         \cfrac{\partial}{\partial x} & \cfrac{\partial}{\partial y} & \cfrac{\partial}{\partial z} \\
         A_x & A_y & A_z
       \end{vmatrix} \\
    &= \nabla \times \mathbf{A}
\end{align}</math>

Many symbolic operations of derivatives can be generalized in a straightforward manner by the gradient operator in Cartesian coordinates. For example, the single-variable [[product rule]] has a direct analogue in the multiplication of scalar fields by applying the gradient operator, as in

:<math>(f g)' = f' g+f g' ~~~ \Longrightarrow ~~~ \nabla(\phi \psi) = (\nabla \phi) \psi + \phi (\nabla \psi).</math>

Many other rules from single variable calculus have [[Vector calculus identities#First derivative identities|vector calculus analogues]] for the gradient, divergence, curl, and Laplacian.

Further notations have been developed for more exotic types of spaces. For calculations in [[Minkowski space]], the [[d'Alembert operator]], also called the d'Alembertian, wave operator, or box operator is represented as <math>\Box</math>, or as <math>\Delta</math> when not in conflict with the symbol for the Laplacian.

==See also==

* {{annotated link|Analytical Society}}
* {{annotated link|Derivative}}
* {{annotated link|Fluxion}}
* {{annotated link|Hessian matrix}}
* {{annotated link|Jacobian matrix}}
* {{annotated link|List of mathematical symbols by subject}}
* [[Operational calculus]]

==References==
{{reflist|2|refs=

<ref name="vpr">{{cite book
 | last1 = Varberg | first1 = Dale E.
 | last2 = Purcell | first2 = Edwin J.
 | last3 = Rigdon | first3 = Steven E.
 | title = Calculus
 | year = 2007
 | publisher = [[Pearson Prentice Hall]]
 | edition = 9th
 | page = 104
 | url = https://archive.org/details/matematika-a-purcell-calculus-9th-ed/page/104
 | isbn = 978-0131469686
}}</ref>

}}

==External links==
* [http://jeff560.tripod.com/calculus.html Earliest Uses of Symbols of Calculus], maintained by Jeff Miller ({{Webarchive|url=https://web.archive.org/web/20200726201326/http://jeff560.tripod.com/calculus.html |date=2020-07-26}}).

{{Differential equations topics}}

[[Category:Differential calculus]]
[[Category:Mathematical notation]]