Leibniz's Rule in One Variable/Also presented as

Leibniz's Rule in One Variable: Also presented as

Leibniz's Rule in One Variable can often be seen presented in this format:

\(\ds \map {\dfrac {\d^n} {\d x^n} } {f g}\)

\(=\)

\(\ds \sum_{k \mathop = 0}^n \dbinom n k \dfrac {\d^k f} {\d x^k} \dfrac {\d^{n - k} g} {\d x^{n - k} }\)

\(\ds \)

\(=\)

\(\ds \dfrac {\d^n f} {\d x^n} g + \dfrac {\d^{n - 1} f} {\d x^{n - 1} } \dfrac {\d g} {\d x} + \dfrac {\d^{n - 2} f} {\d x^{n - 2} } \dfrac {\d^2 g} {\d x^2} + \cdots + \dfrac {\d^{n - k} f} {\d x^{n - k} } \dfrac {\d^k g} {\d^k x} + \cdots + \dfrac {\d f} {\d x} \dfrac {\d^{n - 1} g} {\d^{n - 1} x} + f \dfrac {\d^n g} {\d x^n}\)

Source of Name

This entry was named for Gottfried Wilhelm von Leibniz.

Sources

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.3$ Rules for Differentiation and Integration: Leibniz's Theorem for Differentiation of a Product: $3.3.8$