Notation for differentiation

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 (and its opposite operation, the antidifferentiation or indefinite integration) are listed below.

Leibniz's notation

dy/dx d²y/dx²

Main article: Leibniz's notation

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 ( dy/dx ) can be written as:

$\frac{dy}{dx}$

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

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

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

Higher derivatives ( d²y/dx² ··· dⁿy/dxⁿ ) are expressed as

$\frac{d^ny}{dx^n},\quad\frac{d^n f(x)}{dx^n},\text{ or }\frac{d^n}{dx^n} f(x)$

for the n^th derivative of y = f(x). Historically, this came from the fact that, for example, the third derivative ( d³y/dx³ ) is:

$\frac{d \Bigl(\frac{d \left( \frac{d y} {dx}\right)} {dx}\Bigr)} {dx} = \left(\frac{d}{dx}\right)^3 f(x)$

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

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

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:

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

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:

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

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.

Leibniz's notation for antidifferentiation

∫y dx ∫∫y dx²

For functions of 2 or more variables, see Multiple integral.

Leibniz introduced the integral symbol ( ∫ ) denoted as the current mathematical notation for the integration ( ∫ y dx ) in Analyseos tetragonisticae pars secunda and Methodi tangentium inversae exempla (both from 1675):

$\begin{align} \int y'\,dx &= \int f'(x)\,dx = f(x) + C_0 = y + C_0 \\ \int y\,dx &= \int f(x)\,dx = F(x) + C_1 \\ \int \int y\,dx^2 &= \int \left ( \int y\,dx \right ) dx = \int_{X\times X} f(x)\,dx = \int F(x)\,dx = g(x) + C_2 \\ \underbrace{\int \dots \int}_n y\,\underbrace{dx \dots dx}_n &= \int_{\underbrace{X\times\cdots\times X}_n} f(x)\,dx = \int s(x)\,dx = S(x) + C_n \end{align}$

In order to calculate the n^th antiderivative, Cauchy formula for repeated integration may be of help.

Lagrange's notation

f ʹ(x) f ʺ(x)

One of the most common modern notations for differentiation is due to Joseph Louis Lagrange, defined as φ′(x)

$\phi'(x) \equiv \frac{d\phi (x)}{dx}\,, \quad \text{where}\ \phi(x)\ \text{is an ordinary function of }x\,.$

It uses the prime mark ( f′ f″ f‴ ... f ⁽ⁿ⁾ ) suffixed to a function: the first three derivatives of f are denoted

$f'\;$ for the first derivative,

$f''\;$ for the second derivative,

$f'''\;$ 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 n^th derivative of f is denoted f ⁽ⁿ⁾.

When there are two independent variables for a function f(x,y), the following convention may be followed:

$\begin{align} f^\prime &= \frac{df}{dx} = f_x \\ f_\prime &= \frac{df}{dy} = f_y \\ f^{\prime\prime} &= \frac{d^2 f}{dx^2} = f_{xx} \\ f_\prime^\prime &= \frac{\partial ^2 f}{\partial x \partial y}\ = f_{xy} \\ f_{\prime\prime} &= \frac{d^2 f}{dy^2} = f_{yy} \,, \end{align}$

Lagrange's notation for antidifferentiation

f⁽⁻¹⁾(x) f⁽⁻²⁾(x)

When taking the antiderivative, Lagrange followed Leibniz's notation:

$\phi(x) = \int \phi'(x)\,dx = \int y\,dx \,.$

However, Lagrange's notation for higher order operations extends to integration ( f ⁽⁻¹⁾ f ⁽⁻²⁾ f ⁽⁻³⁾ ... f ⁽⁻ⁿ⁾ ) as:

$f^{(-1)}\;$ for the first integral (do not confuse with the reciprocal function f ⁻¹),

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

$f^{(-3)}\;$ for the third integral.

$f^{(-n)}\;$ for the n^th integral.

Euler's notation

D_xy D²f

Leonhard Euler's notation uses a differential operator suggested by Louis François Antoine Arbogast, denoted as D (D operator) or D̃ (Newton-Leibniz operator)

$D \equiv \frac{d}{dx} \,,$

which must be prefixed to an expression ( D y D² y ... Dⁿ y ) so that the derivatives of a function f are denoted by

$Df \;$ for the first derivative,

$D^2f \;$ for the second derivative,

$D^3f \;$ for the third derivative, and

$D^nf \;$ for the n^th 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 ( D
_x y D²
_x y ... Dⁿ
_x y ), leading to the alternative notation

$D_x y \;$ for the first derivative,

$D^2_x y\;$ for the second derivative,

$D^3_x y \;$ for the third derivative, and

$D^n_x y \;$ for the n^th derivative, for any positive integer n.

If there is only one independent variable present, the subscript to the operator is usually dropped, however. For multiples variables ( D²
_xy y D³
_xyz y ... )

$D^2_{xy} f \equiv \frac{\partial ^2 f}{\partial x \partial y} = f_{xy} \;,$

See § Partial derivatives.

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.

Euler's notation for antidifferentiation

D⁻¹
_xy D⁻²f

Euler's notation ( D⁻¹ y D⁻² y ... D⁻ⁿ y ) is used for the repeated integral as follows

$D^{-1}f(x) \;$ for the first integral,

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

and for the n^th integral (for any positive integer n)

$D^{-n}f(x) = \underbrace{\int \dots \int_0^x}_n f(x)\,\underbrace{dx \dots dx}_n \;.$

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 ( ẏ )

$\dot{y} \equiv \frac{dy}{dt} = \frac{d}{dt}\Bigl(f(t)\Bigr) = D_t y = f'(t) = y'_t$

where the independent variable normally denotes time. Higher order derivatives ( ÿ ÿ̇ ÿ̈ ÿ̈̇ 6ẏ 7ẏ ... 10ẏ ... nẏ ) are shown below:

$\begin{align} \ddot{y} &\equiv \frac{d^2y}{dt^2} = \frac{d}{dt}\left(\frac{dy}{dt}\right) = \frac{d}{dt}\Bigl(\dot{y}\Bigr) = \frac{d}{dt}\Bigl(f'(t)\Bigr) = D_t^2 y = f''(t) = y''_t \\ \overset{...}{y} &= \dot{\ddot{y}} \equiv \frac{d^3y}{dt^3} = D_t^3 y = f'''(t) = y'''_t \\ \overset{\,4}{\dot{y}} &= \overset{....}{y} = \ddot{\ddot{y}} \equiv \frac{d^4y}{dt^4} = D_t^4 y = f^{IV}(t) = y^{(4)}_t \\ \overset{\,5}{\dot{y}} &= \ddot{\overset{...}{y}} = \dot{\ddot{\ddot{y}}} = \ddot{\dot{\ddot{y}}} \equiv \frac{d^5y}{dt^5} = D_t^5 y = f^{V}(t) = y^{(5)}_t \\ \overset{\,6}{\dot{y}} &= \overset{...}{\overset{...}{y}} \equiv \frac{d^6y}{dt^6} = D_t^6 y = f^{VI}(t) = y^{(6)}_t \\ \overset{\,7}{\dot{y}} &= \dot{\overset{...}{\overset{...}{y}}} \equiv \frac{d^7y}{dt^7} = D_t^7 y = f^{VII}(t) = y^{(7)}_t \\ \overset{\,10}{\dot{y}} &= \ddot{\ddot{\ddot{\ddot{\ddot{y}}}}} \equiv \frac{d^{10}y}{dt^{10}} = D_t^{10} y = f^{X}(t) = y^{(10)}_t \\ \overset{\,n}{\dot{y}} &\equiv \frac{d^ny}{dt^n} = D_t^n y = f^{(n)}(t) = y^{(n)}_t \end{align}$

It's often used in classical mechanics and mathematics for time derivatives such as velocity ( ẋ )

$\dot{x} \equiv \frac{dx}{dt} = x'(t) = x'_t \quad \rightsquigarrow \quad x'\ \text{ or }\ v\,,$

acceleration ( ẍ )

$\ddot{x} \equiv \frac{d^2x}{dt^2} = \frac{d}{dt}\Bigl(\dot{x}\Bigr) = x''(t) = x''_t \quad \rightsquigarrow \quad x''\ \text{ or }\ a\,,$

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.

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

When taking the derivative of a dependent variable y = f(x), an alternative notation exists:

$\frac{\dot{y}}{\dot{x}} = \dot{y}:\dot{x} \equiv \frac{dy}{dt}:\frac{dx}{dt} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{dy}{dx} = \frac{d}{dt}\Bigl(f(x)\Bigr) = D y = f'(x) = y'$

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

$\begin{align} \mathcal{X} \ &=\ f(x,y) \,, \\ \cdot\mathcal{X} \ &=\ x\frac{\partial f}{\partial x} = xf_x\,, \\ \mathcal{X}\cdot \ &=\ y\frac{\partial f}{\partial y} = yf_y\,, \\ \colon\mathcal{X}\,\text{ or }\,\cdot\colon\mathcal{X} \ &=\ x^2\frac{\partial^2 f}{\partial x^2} = x^2 f_{xx}\,, \\ \mathcal{X}\colon\,\text{ or }\,\cdot\mathcal{X}\cdot \ &=\ y^2\frac{\partial^2 f}{\partial y^2} = y^2 f_{yy}\,, \\ \cdot\mathcal{X}\cdot\,\text{ or }\,\mathcal{X}\colon\cdot \ &=\ xy\frac{\partial^2 f}{\partial x\partial y} = xy f_{xy}\,, \end{align}$

Newton's notation for antidifferentiation

x̍ x̎

Newton developed many different notations for integration in his Quadratura curvarum (1704) and later 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 the fluent or [time] integral (absement)

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

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

$\overset{\,\prime\prime}{y} = \Box \overset{\,\prime}{y} \equiv \int \overset{\,\prime}{y} \,dt = \int F(t) \,dt = D_t^{-2} y = g(t) + C_2$

and higher order time integrals ( y̎̍ y̎̎ ... ny̍ ) as follows:

$\begin{align} \overset{\,\prime\prime\prime}{y} &= \Box \overset{\,\prime\prime}{y} \equiv \int \overset{\,\prime\prime}{y} \,dt = \int g(t) \,dt = D_t^{-3} y = G(t) + C_3 \\ \overset{\,\prime\prime\prime\prime}{y} &= \Box \overset{\,\prime\prime\prime}{y} \equiv \int \overset{\,\prime\prime\prime}{y} \,dt = \int G(t) \,dt = D_t^{-4} y = h(t) + C_4 \\ \overset{\;n}\overset{\,\prime}{y} &= \Box \overset{\;n-1}\overset{\,\prime}y \equiv \int \overset{\;n-1}\overset{\,\prime}y \,dt = \int s(t) \,dt = D_t^{-n} y = S(t) + C_n \end{align}$

This mathematical notation didn't became widespread because of printing difficulties and other controversies.

Partial derivatives

f_x f_xy

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:

$\begin{align} f_x &= \frac{df}{dx} \\ f_{x x} &= \frac{d^2f}{dx^2}. \end{align}$

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:

$\frac{\partial f}{\partial x} = f_x = \partial_x f = \partial^x f,$

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 $\left(\frac{\partial T}{\partial V}\right)_S$ is the derivative of the temperature T with respect to the volume V while keeping constant the entropy S, while $\left(\frac{\partial T}{\partial V}\right)_P$ 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 $\mathbf{A} = (\mathbf{A}_x, \mathbf{A}_y, \mathbf{A}_z)$, and a scalar field $\varphi$ is $\varphi = f(x,y,z)\,$.

First, the differential operator introduced by William Rowan Hamilton, written ∇ and called called del or nabla, is symbolically defined in the form of a vector,

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

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

∇φ

• Gradient: The gradient $\mathrm{grad\,} \varphi\,$ of the scalar field $\varphi$ is a vector, which is symbolically expressed by the multiplication of ∇ and scalar field $\varphi$,

$\begin{align} \operatorname{grad} \varphi &= \left( \frac{\partial \varphi}{\partial x}, \frac{\partial \varphi}{\partial y}, \frac{\partial \varphi}{\partial z} \right) \\ &= \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right) \varphi \\ &= \nabla \varphi \end{align}$

∇∙A

• Divergence: The divergence $\mathrm{div}\,\mathbf{A}\,$ of the vector field A is a scalar, which is symbolically expressed by the dot product of ∇ and the vector A,

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

∇²φ

• Laplacian: The Laplacian $\operatorname{div} \operatorname{grad} \varphi$ of the scalar field $\varphi$ is a scalar, which is symbolically expressed by the scalar multiplication of ∇² and the scalar field φ,

$\begin{align} \operatorname{div} \operatorname{grad} \varphi &= \nabla \cdot (\nabla \varphi) \\ &= (\nabla \cdot \nabla) \varphi \\ &= \nabla^2 \varphi \\ &= \Delta \varphi \\ \end{align}$

∇×A

• Rotation: The rotation $\mathrm{curl}\,\mathbf{A}\,$, or $\mathrm{rot}\,\mathbf{A}\,$, of the vector field A is a vector, which is symbolically expressed by the cross product of ∇ and the vector A,

$\begin{align} \operatorname{curl} \mathbf{A} &= \left( {\partial A_z \over {\partial y} } - {\partial A_y \over {\partial z} }, {\partial A_x \over {\partial z} } - {\partial A_z \over {\partial x} }, {\partial A_y \over {\partial x} } - {\partial A_x \over {\partial y} } \right) \\ &= \left( {\partial A_z \over {\partial y} } - {\partial A_y \over {\partial z} } \right) \mathbf{i} + \left( {\partial A_x \over {\partial z} } - {\partial A_z \over {\partial x} } \right) \mathbf{j} + \left( {\partial A_y \over {\partial x} } - {\partial A_x \over {\partial y} } \right) \mathbf{k} \\ &= \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\[5pt] \cfrac{\partial}{\partial x} & \cfrac{\partial}{\partial y} & \cfrac{\partial}{\partial z} \\[12pt] A_x & A_y & A_z \end{vmatrix} \\ &= \nabla \times \mathbf{A} \end{align}$

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

$(f g)' = f' g+f g' ~~~ \Longrightarrow ~~~ \nabla(\phi \psi) = (\nabla \phi) \psi + \phi (\nabla \psi).$

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

See also

• Analytical Society
• Derivative
• Jacobian matrix
• Hessian matrix

External links

• Earliest Uses of Symbols of Calculus, maintained by Jeff Miller.