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


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


This notation also makes it possible to describe the nth derivative, where n is a variable. This is written


(“Notation for Differentiation” 2022)