Differential operator

The differential operator is a symbol that represents the action of taking a derivative. Common notations include:

${\begin{matrix}{d \over dx}\\\end{matrix}}$

$D$, where the variable one is differentating to is clear, and

$D_{t}$, where the variable is declared explicitly.

First derivatives are signified as above, but when taking higher, n-th derivatives, the following alterations are useful:

${\begin{matrix}{d^{n} \over dx^{n}}\\D^{n}\end{matrix}}$

The differential operator is not always used to signify differentiation. The act of integration is analogous to "backwards differentiation", and thus we signify the act of integration in terms of the differential operator by using a superscript of -1. For example, the following notations are equivalent:

$D^{-1}(x^{2}+3x)=\int (x^{2}+3x)\;dx$

The differential operator is linear, ie

$D(f+g)=(Df)+(Dg)$

$D(af)=a(Df)$

where f and g are functions, and a is a constant.