Notation For Differentiation - Euler's Notation

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

for the first derivative,

for the second derivative, and

for the nth 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, leading to the alternative notation

for the first derivative,

for the second derivative, and

for the nth derivative, for any positive integer n.

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.