One of the most common modern notations for differentiation [Differentiation] is named after Joseph Louis Lagrange [Joseph Lagrange], even though it was actually invented by Euler and just popularized by the former. In Lagrange’s notation, a prime mark denotes a derivative. If f is a function, then its derivative evaluated at x is written
\[f’(x)\]
It first appeared in print in 1749.[1]
Higher derivatives are indicated using additional prime marks, as in f ″ ( x ) {\displaystyle f’’(x)} f’’(x) for the second derivative and f ‴ ( x ) {\displaystyle f’’’(x)} f’’’(x) for the third derivative. The use of repeated prime marks eventually becomes unwieldy. Some authors continue by employing Roman numerals, usually in lower case,[2][3] as in
\[f^{\mathrm {iv} }(x),f^{\mathrm {v} }(x),f^{\mathrm {vi} }(x),\ldots\]
to denote fourth, fifth, sixth, and higher order derivatives. Other authors use Arabic numerals in parentheses, as in
\[f^{(4)}(x),f^{(5)}(x),f^{(6)}(x),\ldots\]
This notation also makes it possible to describe the nth derivative, where n is a variable. This is written
\[f^{(n)}(x)\]
Lagrange's notation
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“Notation for Differentiation.” 2022. Wikipedia, November. https://en.wikipedia.org/w/index.php?title=Notation_for_differentiation&oldid=1120966008#Lagrange’s_notation.