• 107328  Infos

# Notation for differentiation

There is no single uniform '''notation for differentiation''' Instead several different notations have been proposed by different authors 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

### Lagrange's notation

One of the most common modern notations for differentiation is due to Joseph Louis Lagrange and uses the prime mark:

 $f\text{'}\left(x\right) ;$ for the first derivative $f$(x) ; for the second derivative $f\text{'}\left(x\right) ;$ and 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 the fourth derivative of f would be denoted f(4) The latter notation extends readily to any number of derivatives so thta the nth derivative of f is denoted f(n)

### Leibniz's notation

The other common notation is Leibniz's notation for differentiation which is named after Gottfried Leibniz For the function whose value at x is the derivative of f at x we write:
$frac\left\{dleft\left(f\left(x\right)right\right)\right\}\left\{dx\right\}$

With Leibniz's notation we can write the derivative of f at the point a in two different ways:
$frac\left\{dleft\left(f\left(x\right)right\right)\right\}\left\{dx\right\}left\left\{!!frac\left\{\right\}\left\{\right\}\right\}right|_\left\{x=a\right\} = left\left(frac\left\{dleft\left(f\left(x\right)right\right)\right\}\left\{dx\right\}right\right)\left(a\right)$

If the output of f(x) is another variable for example if y=f(x) we can write the derivative as:
$frac\left\{dy\right\}\left\{dx\right\}$

Higher derivatives are expressed as
$frac\left\{d^nleft\left(f\left(x\right)right\right)\right\}\left\{dx^n\right\}$ or $frac\left\{d^ny\right\}\left\{dx^n\right\}$

for the n-th derivative of f(x) or y respectively Historically this came from the fact that for example the 3rd derivative is:
$frac\left\{d left\left(frac\left\{d left\left( frac\left\{d left\left(f\left(x\right)right\right)\right\} \left\{dx\right\}right\right)\right\} \left\{dx\right\}right\right)\right\} \left\{dx\right\}$

which we can loosely write as:
$left\left(frac\left\{d\right\}\left\{dx\right\}right\right)^3 left\left(f\left(x\right)right\right) =$

frac{d^3}{left(dxright)^3} left(f(x)right)
Dropping brackets gives the notation above
Leibniz's notation allows one to specify the variable for differentiation (in the denominator) This is especially relevant for partial differentiation It also makes the chain rule easy to remember:
$frac\left\{dy\right\}\left\{dx\right\} = frac\left\{dy\right\}\left\{du\right\} cdot frac\left\{du\right\}\left\{dx\right\}$

(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 = dxf '(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 for further information)

### Newton's notation

Newton's notation for differentiation (also called the dot notation for differentiation) requires placing a dot over the function name:
$dot\left\{x\right\} = frac\left\{dx\right\}\left\{dt\right\} = x\text{'}\left(t\right)$

$ddot\left\{x\right\} = frac\left\{d^2x\right\}\left\{dt^2\right\} = x$(t)
and so on
Newton's notation is mainly used in mechanics normally for time derivatives such as velocity and acceleration and in ODE theory It is usually only used for first and second derivatives and then only to denote derivatives with respect to time

### Euler's notation

Euler's notation uses a differential operator denoted as
D which is prefixed to the function with the variable as a subscript of the operator:

 $D_x f\left(x\right) ;$ for the first derivative $D_x^2 f\left(x\right) ;$ for the second derivative and $D_x^n f\left(x\right) ;$ for the nth derivative provided n ≥ 2

This notation can also be abbreviated when taking derivatives of expressions that contain a single variable The subscript to the operator is dropped and is assumed to be the only variable present in the expression In the following examples
u represents any expression of a single variable:

 $D u ;$ for the first derivative $D^2 u ;$ for the second derivative and $D^n u ;$ for the nth derivative provided n ≥ 2

Euler's notation is useful for stating and solving linear differential equations