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.
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Leibniz's notation
- See also: 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 can be written as:
- <math>\frac{dy}{dx}.</math>
The function whose value at x is the derivative of f at x is therefore written
- <math>\frac{d\bigl(f(x)\bigr)}{dx}</math> or <math>\frac{d}{dx}\bigl(f(x)\bigr)</math>
(although strictly speaking this denotes the variable value of the derivative function rather than the derivative function itself). Higher derivatives are expressed as
- <math>\frac{d^ny}{dx^n}</math>, <math>\quad\frac{d^n\bigl(f(x)\bigr)}{dx^n}</math> or <math>\frac{d^n}{dx^n}\bigl(f(x)\bigr)</math>
for the nth derivative of y=f(x). Historically, this came from the fact that, for example, the third derivative is:
- <math>\frac{d \Bigl(\frac{d \left( \frac{d y} {dx}\right)} {dx}\Bigr)} {dx} = \left(\frac{d}{dx}\right)^3 \bigl(f(x)\bigr)</math>
which we can loosely write (dropping the brackets in the denominator) as:
- <math> \frac{d^3}{\left(dx\right)^3} \bigl(f(x)\bigr)=\frac{d^3}{dx^3} \bigl(f(x)\bigr)</math>
as above. With Leibniz's notation, the value of derivative of at a point x=a can be written in two different ways:
- <math>\frac{dy}{dx}\left.{\!\!\frac{}{}}\right|_{x=a} = \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 derivatives. It also makes the chain rule easy to remember:
- <math>\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}.</math>
(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 define du by du = dx•f '(x). In non-standard analysis du is defined as an infinitesimal. It is also interpreted as the exterior derivative du of a function u. See differential (infinitesimal) for further information.)
Lagrange's notation
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
| <math>f'\;</math> | for the first derivative, |
| <math>f\;</math> | for the second derivative, |
| <math>f\;</math> | for the third derivative. |
After this, some authors continue by employing Roman numerals such as fIV for the fourth derivative of f, while others put the number of derivatives in brackets, 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
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
| <math>Df \;</math> | for the first derivative, |
| <math>D^2f \;</math> | for the second derivative, and |
| <math>D^nf \;</math> | for the nth derivative, provided n ≥ 2. |
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
| <math>D_x y \;</math> | for the first derivative, |
| <math>D^2_x y\;</math> | for the second derivative, and |
| <math>D^n_x y \;</math> | for the nth derivative, for any n ≥ 2. |
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.
Newton's notation
- See also: Newton's notation
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 and acceleration:
- <math>\dot{y} = \frac{dy}{dt} </math>
- <math>\ddot{y} = \frac{d^2y}{dt^2} </math>
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's notation is mainly used in mechanics 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.
Other notations
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:
- <math>f_x = \frac{dy}{dx} </math>
- <math>f_{x x} = \frac{d^2y}{dx^2} </math>.
This is especially useful for taking partial derivatives of a function of several variables. Partial derivatives will generally be distinguished from ordinary derivatives by replacing the differential operator d with a "<math>\partial</math>" 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:
- <math>\frac{\partial f}{\partial x} = f_x = \partial_x f = \partial^x f </math>,
where the final two notations are equivalent in flat Euclidean Space but are different in other manifolds. For functions of several variables, we can map derivatives into a vector space using the gradient operator, indicated by a nabla (<math>\nabla</math>) as the vector of partial derivatives of the function. So in Cartesian coordinates,
- <math>\nabla f(x,y,z) = \langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \rangle </math>.
Further operations can be defined through divergence and curl, which are the inner product and cross product of the same operator. Specific notations have been developed for particular types of spaces, including the D'Alembert operator (<math>\Delta</math>) or "box" operator (<math>\Box</math>) used in Minkowski space. Other generalizations of the derivative can be found in various subfields of mathematics, physics, and engineering.
