Notation for differentiation

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In differential calculus, there is no single uniform notation for differentiation. Instead, several different notations for the derivative of a function or variable have been proposed by different 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 are listed below.

Leibniz's notation

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dy
dx
dx2

The original notation employed by Gottfried Leibniz is used throughout mathematics. It is particularly common when the equation y = f(x) is regarded as a functional relationship between dependent and independent variables y and x. In this case the derivative can be written as:

${\displaystyle {\frac {dy}{dx}}}$

The function whose value at x is the derivative of f at x is therefore written

${\displaystyle {\frac {d{\bigl (}f(x){\bigr )}}{dx}}{\text{ or }}{\frac {d}{dx}}{\bigl (}f(x){\bigr )}}$

(although strictly speaking this denotes the variable value of the derivative function rather than the derivative function itself).

Higher derivatives are expressed as

${\displaystyle {\frac {d^{n}y}{dx^{n}}},\quad {\frac {d^{n}{\bigl (}f(x){\bigr )}}{dx^{n}}},{\text{ or }}{\frac {d^{n}}{dx^{n}}}{\bigl (}f(x){\bigr )}}$

for the nth derivative of y = f(x). Historically, this came from the fact that, for example, the third derivative is:

${\displaystyle {\frac {d{\Bigl (}{\frac {d\left({\frac {dy}{dx}}\right)}{dx}}{\Bigr )}}{dx}}=\left({\frac {d}{dx}}\right)^{3}{\bigl (}f(x){\bigr )}}$

which we can loosely write (dropping the brackets in the denominator) as:

${\displaystyle {\frac {d^{3}}{\left(dx\right)^{3}}}{\bigl (}f(x){\bigr )}={\frac {d^{3}}{dx^{3}}}{\bigl (}f(x){\bigr )}}$

as above.

With Leibniz's notation, the value of the derivative of y at a point x = a can be written in two different ways:

${\displaystyle {\frac {dy}{dx}}\left.{\!\!{\frac {}{}}}\right|_{x=a}={\frac {dy}{dx}}(a).}$

Leibniz's notation allows one to specify the variable for differentiation (in the denominator). This is especially helpful when considering partial derivatives. It also makes the chain rule easy to remember and recognize:

${\displaystyle {\frac {dy}{dx}}={\frac {dy}{du}}\cdot {\frac {du}{dx}}.}$

In the formulation of calculus in terms of limits, the du symbol has been assigned various meanings by various authors.

Some authors do not assign a meaning to du by itself, but only as part of the symbol du/dx.

Others define dx as an independent variable, and use d(x + y) = dx + dy and d(x·y) = dx·y + x·dy as formal axioms for differentiation. See differential algebra.

In non-standard analysis du is defined as an infinitesimal.

It is also interpreted as the exterior derivative du of a function u.

Lagrange's notation

f ʹ(x)  f ʺ(x)

One of the most common modern notations for differentiation is due to Joseph Louis Lagrange and uses the prime mark: the first three derivatives of f are denoted

${\displaystyle f'\;}$ for the first derivative,
${\displaystyle f''\;}$ for the second derivative,
${\displaystyle f'''\;}$ for the third derivative.

After this, some authors continue by employing Roman numerals such as f IV for the fourth derivative of f, while others put the ordinal of the derivative in parentheses, so that the fourth derivative of f would be denoted f (4). The latter notation extends readily to any number of derivatives, so that the nth derivative of f is denoted f (n).

Euler's notation

DxTemplate:Resizey D2f

Euler's notation uses a differential operator, denoted as D, which is prefixed to the function so that the derivatives of a function f are denoted by

${\displaystyle Df\;}$ for the first derivative,
${\displaystyle D^{2}f\;}$ for the second derivative, and
${\displaystyle D^{n}f\;}$ for the nth derivative, for any positive integer n.

When taking the derivative of a dependent variable y = f(x) it is common to add the independent variable x as a subscript to the D notation, leading to the alternative notation

${\displaystyle D_{x}y\;}$ for the first derivative,
${\displaystyle D_{x}^{2}y\;}$ for the second derivative, and
${\displaystyle D_{x}^{n}y\;}$ for the nth derivative, for any positive integer n.

If there is only one independent variable present, the subscript to the operator is usually dropped, however.

Euler's notation is useful for stating and solving linear differential equations, as it simplifies presentation of the differential equation, which can make seeing the essential elements of the problem easier.

Newton's notation

ẋ ẍ

Newton's notation for differentiation (also called the dot notation for differentiation) requires placing a dot over the dependent variable and is often used for time derivatives such as velocity

${\displaystyle {\dot {y}}={\frac {dy}{dt}}\,,}$
${\displaystyle {\ddot {y}}={\frac {d^{2}y}{dt^{2}}}\,,}$

and so on. It can also be used as a direct substitute for the prime in Lagrange's notation. Again this is common for functions f(t) of time. Newton referred to this as a fluxion.[1]

Newton's notation is mainly used in mechanics, physics, and the theory of ordinary differential equations. It is usually only used for first and second derivatives, and then, only to denote derivatives with respect to time.

Dot notation is not very useful for higher-order derivatives, but in mechanics and other engineering fields, the use of higher than second-order derivatives is limited.

In physics, macroeconomics and other fields, Newton's notation is used mostly for time derivatives, as opposed to slope or position derivatives.

Newton did not develop a standard mathematical notation for integration but used many different notations.

Partial derivatives

fx  fxy

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

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

${\displaystyle f_{x}={\frac {df}{dx}}}$
${\displaystyle f_{xx}={\frac {d^{2}f}{dx^{2}}}.}$

This is especially useful for taking partial derivatives of a function of several variables.

∂f
∂x

Partial derivatives will generally be distinguished from ordinary derivatives by replacing the differential operator d with a "" symbol. For example, we can indicate the partial derivative of f(x,y,z) with respect to x, but not to y or z in several ways:

${\displaystyle {\frac {\partial f}{\partial x}}=f_{x}=\partial _{x}f=\partial ^{x}f,}$

where the final two notations are equivalent in flat Euclidean Space but are different in other manifolds.

Other notations can be found in various subfields of mathematics, physics, and engineering, see for example the Maxwell relations of thermodynamics. The symbol ${\displaystyle \left({\frac {\partial T}{\partial V}}\right)_{S}}$ is the derivative of the temperature T with respect to the volume V while keeping constant the entropy S, while ${\displaystyle \left({\frac {\partial T}{\partial V}}\right)_{P}}$ is the derivative of the temperature with respect to the volume while keeping constant the pressure P.

Notation in vector calculus

Vector calculus concerns differentiation and integration of vector or scalar fields particularly in a three-dimensional Euclidean space, and uses specific notations of differentiation. In a Cartesian coordinate o-xyz, assuming a vector field A is ${\displaystyle \mathbf {A} =(\mathbf {A} _{x},\mathbf {A} _{y},\mathbf {A} _{z})}$, and a scalar field ${\displaystyle \varphi }$ is ${\displaystyle \varphi =f(x,y,z)\,}$.

First, a differential operator, or a Hamilton operator which is called del is symbolically defined in the form of a vector,

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

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

φ
{\displaystyle {\begin{aligned}\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{aligned}}}
∇∙A
{\displaystyle {\begin{aligned}\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{aligned}}}
2φ
{\displaystyle {\begin{aligned}\operatorname {div} \operatorname {grad} \varphi &=\nabla \cdot (\nabla \varphi )\\&=(\nabla \cdot \nabla )\varphi \\&=\nabla ^{2}\varphi \\&=\Delta \varphi \\\end{aligned}}}
where, ${\displaystyle \Delta =\nabla ^{2}}$ is called a Laplacian operator.
∇×A
{\displaystyle {\begin{aligned}\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} \\[5pt]{\cfrac {\partial }{\partial x}}&{\cfrac {\partial }{\partial y}}&{\cfrac {\partial }{\partial z}}\\[12pt]A_{x}&A_{y}&A_{z}\end{vmatrix}}\\&=\nabla \times \mathbf {A} \end{aligned}}}

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

${\displaystyle (fg)'=f'g+fg'~~~\Longrightarrow ~~~\nabla (\phi \psi )=(\nabla \phi )\psi +\phi (\nabla \psi ).}$

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 ${\displaystyle \Box }$, or as ${\displaystyle \Delta }$ when not in conflict with the symbol for the Laplacian.