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Indifferential calculus, there is no single standardnotation for differentiation. Instead, several notations for thederivative of afunction or adependent variable have been proposed by various mathematicians, includingLeibniz,Newton,Lagrange, andArbogast. The usefulness of each notation depends on the context in which it is used, and it is sometimes advantageous to use more than one notation in a given context. For more specialized settings—such aspartial derivatives inmultivariable calculus,tensor analysis, orvector calculus—other notations, such as subscript notation or the∇ operator are common. The most common notations for differentiation (and its opposite operation,antidifferentiation orindefinite integration) are listed below.

The original notation employed byGottfried Leibniz is used throughout mathematics. It is particularly common when the equationy =f(x) is regarded as a functional relationship betweendependent and independent variablesy andx. Leibniz's notation makes this relationship explicit by writing the derivative as:^[1]$${\displaystyle \frac{dy}{dx}.}$$Furthermore, the derivative off atx is therefore written$${\displaystyle \frac{df}{dx}(x)\text{ or }\frac{df(x)}{dx}\text{ or }\frac{d}{dx}f(x).}$$

Higher derivatives are written as:^[2]$${\displaystyle \frac{d^{2}y}{d{x}^2},\frac{d^{3}y}{d{x}^3},\frac{d^{4}y}{d{x}^4},\dots ,\frac{d^{n}y}{d{x}^n}.}$$This is a suggestive notational device that comes from formal manipulations of symbols, as in,$${\displaystyle \frac{d\left(\frac{dy}{dx}\right)}{dx}={\left(\frac{d}{dx}\right)}^{2}y=\frac{d^{2}y}{d{x}^2}.}$$

The value of the derivative ofy at a pointx =a may be expressed in two ways using Leibniz's notation:$${\displaystyle {\frac{dy}{dx}|}_{x=a}\text{ or }\frac{dy}{dx}(a).}$$

Leibniz's notation allows one to specify the variable for differentiation (in the denominator). This is especially helpful when consideringpartial derivatives. It also makes thechain rule easy to remember and recognize:$${\displaystyle \frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}.}$$

Leibniz's notation for differentiation does not require assigning meaning to symbols such asdx ordy (known asdifferentials) on their own, and some authors do not attempt to assign these symbols meaning.^[1] Leibniz treated these symbols asinfinitesimals. Later authors have assigned them other meanings, such as infinitesimals innon-standard analysis, orexterior derivatives. Commonly,dx is left undefined or equated with${\displaystyle \mathrm{\Delta }x}$, whiledy is assigned a meaning in terms ofdx, via the equation

${\displaystyle dy=\frac{dy}{dx}\cdot dx,}$

which may also be written, e.g.

${\displaystyle df=f^{\prime }(x)\cdot dx}$

(seebelow). Such equations give rise to the terminology found in some texts wherein the derivative is referred to as the "differential coefficient" (i.e., thecoefficient ofdx).

Some authors and journals set the differential symbold inroman type instead ofitalic:dx. TheISO/IEC 80000 scientific style guide recommends this style.

One of the most common modern notations for differentiation is named afterJoseph Louis Lagrange, although it was in fact invented byEuler and popularized by the former. In Lagrange's notation, aprime mark denotes a derivative – hence it is sometimes calledprime notation. Iff is a function, then its derivative evaluated atx is written

${\displaystyle f^{\prime }(x)}$.

It first appeared in print in 1749.^[3]

Higher derivatives are indicated using additional prime marks, as in${\displaystyle f^{″}(x)}$ for thesecond derivative and${\displaystyle f^{‴}(x)}$ for thethird derivative. The use of repeated prime marks eventually becomes unwieldy; some authors continue by employingRoman numerals, usually in lower case,^[4][5] as in

${\displaystyle f^{\mathrm{i}\mathrm{v}}(x),f^{\mathrm{v}}(x),f^{\mathrm{v}\mathrm{i}}(x),\dots ,}$

to denote fourth, fifth, sixth, and higher order derivatives. Other authors use Arabic numerals in parentheses, as in

${\displaystyle f^{(4)}(x),f^{(5)}(x),f^{(6)}(x),\dots .}$

This notation also makes it possible to describe thenth derivative, wheren is a variable. This is written

${\displaystyle f^{(n)}(x).}$

Unicode characters related to Lagrange's notation include

• U+2032 ◌′PRIME (derivative)
• U+2033 ◌″DOUBLE PRIME (double derivative)
• U+2034 ◌‴TRIPLE PRIME (third derivative)
• U+2057 ◌⁗QUADRUPLE PRIME (fourth derivative)

When there are two independent variables for a function${\displaystyle f(x,y)}$, the following notation was sometimes used:^[6]

When taking the antiderivative, Lagrange followed Leibniz's notation:^[7]

${\displaystyle f(x)=\int f^{\prime }(x)dx=\int y^{\prime }dx.}$

However, because integration is the inverse operation of differentiation, Lagrange's notation for higher order derivatives extends to integrals as well. Repeated integrals off may be written as

${\displaystyle f^{(-1)}(x)}$ for the first integral (this is easily confused with theinverse function${\displaystyle f^{-1}(x)}$),

${\displaystyle f^{(-2)}(x)}$ for the second integral,

${\displaystyle f^{(-3)}(x)}$ for the third integral, and

${\displaystyle f^{(-n)}(x)}$ for thenth integral.

This notation is sometimes calledEuler's notation although it was introduced byLouis François Antoine Arbogast,^[8] and it seems thatLeonhard Euler did not use it.^{[citation needed]}

This notation uses adifferential operator denoted asD (D operator)^{[9][failed verification]} orD̃ (Newton–Leibniz operator).^[10] When applied to a functionf(x), it is defined by

${\displaystyle (Df)(x)=\frac{df(x)}{dx}.}$

Higher derivatives are notated as "powers" ofD (where the superscripts denote iteratedcomposition ofD), as in^[6]

${\displaystyle D^{2}f}$ for the second derivative,

${\displaystyle D^{3}f}$ for the third derivative, and

${\displaystyle D^{n}f}$ for thenth 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: iff is a function of a variablex, this is done by writing^[6]

${\displaystyle D_{x}f}$ for the first derivative,

${\displaystyle D_x^{2}f}$ for the second derivative,

${\displaystyle D_x^{3}f}$ for the third derivative, and

${\displaystyle D_x^{n}f}$ for thenth derivative.

Whenf is a function of several variables, it is common to use "∂", a stylized cursive lower-case d, rather than "D". As above, the subscripts denote the derivatives that are being taken. For example, the second partial derivatives of a function${\displaystyle f(x,y)}$ are:^[6]

See§ Partial derivatives.

D-notation is useful in the study ofdifferential equations and indifferential algebra.

D-notation can be used for antiderivatives in the same way that Lagrange's notation is^[11] as follows^[10]

${\displaystyle D^{-1}f(x)}$ for a first antiderivative,

${\displaystyle D^{-2}f(x)}$ for a second antiderivative, and

${\displaystyle D^{-n}f(x)}$ for annth antiderivative.

Isaac Newton's notation for differentiation (also called thedot notation,fluxions, or sometimes, crudely, theflyspeck notation^[12] for differentiation) places a dot over the dependent variable. That is, ify is a function oft, then the derivative ofy with respect tot is

${\displaystyle \dot{y}}$

Higher derivatives are represented using multiple dots, as in

${\displaystyle \ddot{y},\stackrel{...}{y}}$

Newton extended this idea quite far:^[13]

Unicode characters related to Newton's notation include:

• U+0307 ◌̇COMBINING DOT ABOVE (derivative)
• U+0308 ◌̈COMBINING DIAERESIS (double derivative)
• U+20DB ◌⃛COMBINING THREE DOTS ABOVE (third derivative) ← replaced by "combining diaeresis" + "combining dot above".
• U+20DC ◌⃜COMBINING FOUR DOTS ABOVE (fourth derivative) ← replaced by "combining diaeresis" twice.
• U+030D ◌̍COMBINING VERTICAL LINE ABOVE (integral)
• U+030E ◌̎COMBINING DOUBLE VERTICAL LINE ABOVE (second integral)
• U+25AD ▭WHITE RECTANGLE (integral)
• U+20DE ◌⃞COMBINING ENCLOSING SQUARE (integral)
• U+1DE0 ◌ᷠCOMBINING LATIN SMALL LETTER N (nth derivative)

Newton's notation is generally used when the independent variable denotestime. If locationy is a function oft, then${\displaystyle \dot{y}}$ denotesvelocity^[14] and${\displaystyle \ddot{y}}$ denotesacceleration.^[15] This notation is popular inphysics andmathematical physics. It also appears in areas of mathematics connected with physics such asdifferential equations.

When taking the derivative of a dependent variabley =f(x), an alternative notation exists:^[16]

${\displaystyle \frac{\dot{y}}{\dot{x}}=\dot{y}:\dot{x}\equiv \frac{dy}{dt}:\frac{dx}{dt}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{dy}{dx}=\frac{d}{dx}(f(x))=Dy=f^{\prime }(x)=y^{\prime }.}$

Newton developed the following partial differential operators using side-dots on a curved X ( ⵋ ). Definitions given by Whiteside are below:^[17][18]

Newton developed many different notations forintegration in hisQuadratura curvarum (1704) andlater works: he wrote a small vertical bar or prime above the dependent variable (y̍ ), a prefixing rectangle (▭y), or the inclosure of the term in a rectangle (y) to denote thefluent or time integral (absement).

To denote multiple integrals, Newton used two small vertical bars or primes (y̎), or a combination of previous symbols▭y̍ y̍, to denote the second time integral (absity).

${\displaystyle \stackrel{\mathrm{\prime }\mathrm{\prime }}{y}=◻\stackrel{\mathrm{\prime }}{y}\equiv \int \stackrel{\mathrm{\prime }}{y}dt=\int F(t)dt=D_t^{-2}y=g(t)+C_2}$

Higher order time integrals were as follows:^[19]

Thismathematical notation did not become widespread because of printing difficulties^{[Citation needed]} and theLeibniz–Newton calculus controversy.

When more specific types of differentiation are necessary, such as inmultivariate calculus ortensor analysis, other notations are common.

For a functionf of a single independent variablex, we can express the derivative using subscripts of the independent variable:

This type of notation is especially useful for takingpartial derivatives of a function of several variables.

Partial derivatives are generally distinguished from ordinary derivatives by replacing the differential operatord with a "∂" symbol. For example, we can indicate the partial derivative off(x, y, z) with respect tox, but not toy orz in several ways:

${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }x}=f_x={\mathrm{\partial }}_{x}f.}$

What makes this distinction important is that a non-partial derivative such as${\displaystyle \frac{df}{dx}}$may, depending on the context, be interpreted as a rate of change in${\displaystyle f}$ relative to${\displaystyle x}$ when all variables are allowed to vary simultaneously, whereas with a partial derivative such as${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }x}}$ 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 theMaxwell relations ofthermodynamics. The symbol${\displaystyle {\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }V}\right)}_S}$ is the derivative of the temperatureT with respect to the volumeV while keeping constant the entropy (subscript)S, while${\displaystyle {\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }V}\right)}_P}$ is the derivative of the temperature with respect to the volume while keeping constant the pressureP. 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

and so on. Mixed partial derivatives can be expressed as

${\displaystyle \frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }y\mathrm{\partial }x}=f_{xy}.}$

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

So-calledmulti-index notation is used in situations when the above notation becomes cumbersome or insufficiently expressive. When considering functions on${\displaystyle {\mathbb{R}}^n}$, we define a multi-index to be an ordered list of${\displaystyle n}$ non-negative integers:${\displaystyle \alpha =({\alpha }_1,\dots ,{\alpha }_n),{\alpha }_i\in {\mathbb{Z}}_{\ge 0}}$. We then define, for${\displaystyle f:{\mathbb{R}}^n\to X}$, the notation

${\displaystyle {\mathrm{\partial }}^{\alpha }f=\frac{{\mathrm{\partial }}^{{\alpha }_1}}{\mathrm{\partial }{x}_1^{{\alpha }_1}}\cdots \frac{{\mathrm{\partial }}^{{\alpha }_n}}{\mathrm{\partial }{x}_n^{{\alpha }_n}}f}$

In this way some results (such as theLeibniz rule) that are tedious to write in other ways can be expressed succinctly -- some examples can be found in thearticle on multi-indices.^[20]

Vector calculus concernsdifferentiation andintegration ofvector orscalar fields. Several notations specific to the case of three-dimensionalEuclidean space are common.

Assume that(x,y,z) is a givenCartesian coordinate system, thatA is avector field with components${\displaystyle \mathbf{A}=(A_x,A_y,A_z)}$, and that${\displaystyle \phi =\phi (x,y,z)}$ is ascalar field.

The differential operator introduced byWilliam Rowan Hamilton, written∇ and calleddel or nabla, is symbolically defined in the form of a vector,

${\displaystyle \mathrm{\nabla }=\left(\frac{\mathrm{\partial }}{\mathrm{\partial }x},\frac{\mathrm{\partial }}{\mathrm{\partial }y},\frac{\mathrm{\partial }}{\mathrm{\partial }z}\right),}$

where the terminologysymbolically reflects that the operator ∇ will also be treated as an ordinary vector.

• Gradient: The gradient${\displaystyle \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\phi }$ of the scalar field${\displaystyle \phi }$ is a vector, which is symbolically expressed by themultiplication of ∇ and scalar field${\displaystyle \phi }$,

• Divergence: The divergence${\displaystyle \mathrm{d}\mathrm{i}\mathrm{v}\mathbf{A}}$ of the vector fieldA is a scalar, which is symbolically expressed by thedot product of ∇ and the vectorA,

• Laplacian: The Laplacian${\displaystyle \mathrm{div}\mathrm{grad}\phi }$ of the scalar field${\displaystyle \phi }$ is a scalar, which is symbolically expressed by the scalar multiplication of ∇² and the scalar fieldφ,

• Rotation: The rotation${\displaystyle \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\mathbf{A}}$, or${\displaystyle \mathrm{r}\mathrm{o}\mathrm{t}\mathbf{A}}$, of the vector fieldA is a vector, which is symbolically expressed by thecross product of ∇ and the vectorA,

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

${\displaystyle (fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }⟹\mathrm{\nabla }(\varphi \psi )=(\mathrm{\nabla }\varphi )\psi +\varphi (\mathrm{\nabla }\psi ).}$

Many other rules from single variable calculus havevector calculus analogues for the gradient, divergence, curl, and Laplacian.

Further notations have been developed for more exotic types of spaces. For calculations inMinkowski space, thed'Alembert operator, also called the d'Alembertian, wave operator, or box operator is represented as${\displaystyle ◻}$, or as${\displaystyle \mathrm{\Delta }}$ when not in conflict with the symbol for the Laplacian.

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