Notation for differentiation

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. Leibniz's notation makes this relationship explicit by writing the derivative as

d y d x .

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

d f d x ( x )  or  d f ( x ) d x  or  d d x f ( x ) .

Higher derivatives are written as

d 2 y d x 2 , d 3 y d x 3 , d 4 y d x 4 , … , d n y d x n .

This is a suggestive notational device that comes from formal manipulations of symbols, as in,

d ( d ( d y d x ) d x ) d x = ( d d x ) 3 y = d 3 y d x 3 .

Logically speaking, these equalities are not theorems. Instead, they are simply definitions of notation.

The value of the derivative of y at a point x = a may be expressed in two ways using Leibniz's notation:

d y d x | x = a = d y d x ( 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:

d y d x = d y d u ⋅ d u d x .

Leibniz's notation for differentiation does require assigning a meaning to symbols such as dx or dy on their own, and some authors do not attempt to assign these symbols meaning. Leibniz treated these symbols as infinitesimals. Later authors have assigned them other meanings, such as infinitesimals in non-standard analysis or exterior derivatives.

Leibniz's notation for antidifferentiation

Leibniz introduced the integral symbol ∫ in Analyseos tetragonisticae pars secunda and Methodi tangentium inversae exempla (both from 1675). It is now the standard symbol for integration.

∫ y ′ d x = ∫ f ′ ( x ) d x = f ( x ) + C 0 = y + C 0 ∫ y d x = ∫ f ( x ) d x = F ( x ) + C 1 ∫ ∫ y d x 2 = ∫ ( ∫ y d x ) d x = ∫ X × X f ( x ) d x = ∫ F ( x ) d x = g ( x ) + C 2 ∫ … ∫ ⏟ n y d x … d x ⏟ n = ∫ X × ⋯ × X ⏟ n f ( x ) d x = ∫ s ( x ) d x = S ( x ) + C n

Lagrange's notation

One of the most common modern notations for differentiation is due to Joseph Louis Lagrange. In Lagrange's notation, a prime mark denotes a derivative. If f is a function, then its derivative is written

f ′ ( x ) .

Lagrange first used the notation in unpublished works, and it appeared in print in 1770.

Higher derivatives are indicated using additional prime marks, as in f ″ ( x ) for the second derivative and f ‴ ( x ) for the third derivative. The use of repeated prime marks eventually becomes unwieldy. Some authors continue by employing Roman numerals, as in

f I V ( x ) , f V ( x ) , f V I ( x ) , … ,

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

f ( 4 ) ( x ) , f ( 5 ) ( x ) , f ( 6 ) ( x ) , … .

This notation also makes it possible to describe the nth derivative, where n is a variable. This is written

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 f(x,y), the following convention may be followed:

f ′ = d f d x = f x f ′ = d f d y = f y f ′ ′ = d 2 f d x 2 = f x x f ′ ′ = ∂ 2 f ∂ x ∂ y   = f x y f ′ ′ = d 2 f d y 2 = f y y ,

Lagrange's notation for antidifferentiation

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

f ( x ) = ∫ f ′ ( x ) d x = ∫ y d x .

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

f ( − 1 ) ( x ) for the first integral (this is easily confused with the inverse function f − 1 ( x ) ), f ( − 2 ) ( x ) for the second integral, f ( − 3 ) ( x ) for the third integral, and f ( − n ) ( x ) for the nth integral.

Euler's notation

Leonhard Euler's notation uses a differential operator suggested by Louis François Antoine Arbogast, denoted as D (D operator) or D̃ (Newton-Leibniz operator) When applied to a function f(x), it is defined by

D f = d f d x .

Higher derivatives are notated as powers of D, as in

D 2 f for the second derivative, D 3 f for the third derivative, and D n f for the nth derivative.

Euler's notation leaves implicit the variable with respect to which differentiation is being done. However, this variable can also be notated explicitly. When f is a function of a variable x, this is done by writing

D x f for the first derivative, D x 2 f for the second derivative, D x 3 f for the third derivative, and D x n f for the nth derivative.

This notation is particularly handy when f is a function of several variables, in which case the subscript denotes the derivatives that are being taken. For example, the second partial derivatives of a function f(x, y) are:

D x x f = ∂ 2 f ∂ x 2 , D x y f = ∂ 2 f ∂ x ∂ y , D y x f = ∂ 2 f ∂ y ∂ x , D y y f = ∂ 2 f ∂ y 2 .

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

Euler's notation can be used for antidifferentiation in the same way that Lagrange's notation is. as follows

D − 1 f ( x ) for a first antiderivative, D − 2 f ( x ) for a second antiderivative, and D − n f ( x ) for an nth antiderivative.

Newton's notation

Newton's notation for differentiation (also called the dot notation for differentiation) places a dot over the dependent variable. That is, if y is a function of x, then the derivative of y with respect to x is

y ˙ .

Higher derivatives are represented using multiple dots, as in

y ¨ , y . . . .

Newton extended this idea quite far:

y ¨ ≡ d 2 y d t 2 = d d t ( d y d t ) = d d t ( y ˙ ) = d d t ( f ′ ( t ) ) = D t 2 y = f ″ ( t ) = y t ″ y . . . = y ¨ ˙ ≡ d 3 y d t 3 = D t 3 y = f ‴ ( t ) = y t ‴ y ˙ 4 = y . . . . = y ¨ ¨ ≡ d 4 y d t 4 = D t 4 y = f I V ( t ) = y t ( 4 ) y ˙ 5 = y . . . ¨ = y ¨ ¨ ˙ = y ¨ ˙ ¨ ≡ d 5 y d t 5 = D t 5 y = f V ( t ) = y t ( 5 ) y ˙ 6 = y . . . . . . ≡ d 6 y d t 6 = D t 6 y = f V I ( t ) = y t ( 6 ) y ˙ 7 = y . . . . . . ˙ ≡ d 7 y d t 7 = D t 7 y = f V I I ( t ) = y t ( 7 ) y ˙ 10 = y ¨ ¨ ¨ ¨ ¨ ≡ d 10 y d t 10 = D t 10 y = f X ( t ) = y t ( 10 ) y ˙ n ≡ d n y d t n = D t n y = f ( n ) ( t ) = y t ( n )

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 (n^th derivative)

Newton's notation is generally used when the independent variables denotes time. If x is a function of t, then x ˙ denotes velocity and x ¨ denotes acceleration. This notation is popular in physics and mathematical physics. It also appears in areas of mathematics connected with physics such as differential equations. It is only popular for first and second derivatives, but in applications these are usually the only derivatives that are necessary.

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

y ˙ x ˙ = y ˙ : x ˙ ≡ d y d t : d x d t = d y d t d x d t = d y d x = d d t ( f ( x ) ) = 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:

X   =   f ( x , y ) , ⋅ X   =   x ∂ f ∂ x = x f x , X ⋅   =   y ∂ f ∂ y = y f y , : X  or  ⋅ : X   =   x 2 ∂ 2 f ∂ x 2 = x 2 f x x , X :  or  ⋅ X ⋅   =   y 2 ∂ 2 f ∂ y 2 = y 2 f y y , ⋅ X ⋅  or  X : ⋅   =   x y ∂ 2 f ∂ x ∂ y = x y f x y ,

Newton's notation for antidifferentiation

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).

y = ◻ y ˙ ≡ ∫ y ˙ d t = ∫ f ′ ( t ) d t = D t − 1 ( D t y ) = f ( t ) + C 0 = y t + C 0 y ′ = ◻ y ≡ ∫ y d t = ∫ f ( t ) d t = D t − 1 y = F ( t ) + C 1

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).

y ′ ′ = ◻ y ′ ≡ ∫ y ′ d t = ∫ F ( t ) d t = D t − 2 y = g ( t ) + C 2

Higher order time integrals were as follows:

y ′ ′ ′ = ◻ y ′ ′ ≡ ∫ y ′ ′ d t = ∫ g ( t ) d t = D t − 3 y = G ( t ) + C 3 y ′ ′ ′ ′ = ◻ y ′ ′ ′ ≡ ∫ y ′ ′ ′ d t = ∫ G ( t ) d t = D t − 4 y = h ( t ) + C 4 y ′ n = ◻ y ′ n − 1 ≡ ∫ y ′ n − 1 d t = ∫ s ( t ) d t = D t − n y = S ( t ) + C n

This mathematical notation didn't become widespread because of printing difficulties and the Leibniz–Newton calculus controversy.

Partial derivatives

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:

f x = d f d x f x x = d 2 f d x 2 .

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

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 f(x,y,z) with respect to x, but not to y or z in several ways:

∂ f ∂ x = f x = ∂ x f .

Other notations can be found in various subfields of mathematics, physics, and engineering, see for example the Maxwell relations of thermodynamics. The symbol ( ∂ T ∂ V ) S is the derivative of the temperature T with respect to the volume V while keeping constant the entropy (subscript) S, while ( ∂ T ∂ V ) 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. Several notations specific to the case of three-dimensional Euclidean space are common.

Assume that (x, y, z) is a given Cartesian coordinate system, that A is a vector field with components A = ( A x , A y , A z ) , and that φ = φ ( x , y , z ) is a scalar field.

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

∇ = ( ∂ ∂ x , ∂ ∂ y , ∂ ∂ z ) ,

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

Gradient: The gradient g r a d φ of the scalar field φ is a vector, which is symbolically expressed by the multiplication of ∇ and scalar field φ ,

Divergence: The divergence d i v A of the vector field A is a scalar, which is symbolically expressed by the dot product of ∇ and the vector A,

Laplacian: The Laplacian div ⁡ grad ⁡ φ of the scalar field φ is a scalar, which is symbolically expressed by the scalar multiplication of ∇² and the scalar field φ,

Rotation: The rotation c u r l A , or r o t A , of the vector field A is a vector, which is symbolically expressed by the cross product of ∇ and the vector A,

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 ′       ⟹       ∇ ( ϕ ψ ) = ( ∇ ϕ ) ψ + ϕ ( ∇ ψ ) .

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 ◻ , or as Δ when not in conflict with the symbol for the Laplacian.