Leibniz's Rule/One Variable/Third Derivative
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Theorem
Let $f$ and $g$ be real functions defined on the open interval $I$.
Let $x \in I$ be a point in $I$ at which both $f$ and $g$ are thrice differentiable.
Then:
- $\paren {\map f x \map g x}''' = \map f x \map {g'''} x + 3 \map {f'} x \map {g''} x + 3 \map {f''} x \map {g'} x + \map {f'''} x \map g x$
Proof
From Leibniz's Rule in One Variable:
- $\ds \paren {\map f x \map g x}^{\paren n} = \sum_{k \mathop = 0}^n \binom n k \map {f^{\paren k} } x \map {g^{\paren {n - k} } } x$
where $\paren n$ denotes the order of the derivative.
Setting $n = 3$:
\(\ds \paren {\map f x \map g x}'''\) | \(=\) | \(\ds \paren {\map f x \map g x}^{\paren 3}\) | Definition of Nth Derivative | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^3 \binom n k \map {f^{\paren k} } x \map {g^{\paren {n - k} } } x\) | Leibniz's Rule in One Variable | |||||||||||
\(\ds \) | \(=\) | \(\ds \binom 3 0 \map {f^{\paren 0} } x \map {g^{\paren 3} } x + \binom 3 1 \map {f^{\paren 1} } x \map {g^{\paren 2} } x + \binom 3 2 \map {f^{\paren 2} } x \map {g^{\paren 1} } x + \binom 3 3 \map {f^{\paren 3} } x \map {g^{\paren 0} } x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {f^{\paren 0} } x \map {g^{\paren 3} } x + 3 \map {f^{\paren 1} } x \map {g^{\paren 2} } x + 3 \map {f^{\paren 2} } x \map {g^{\paren 1} } x + \map {f^{\paren 3} } x \map {g^{\paren 0} } x\) | Definition of Binomial Coefficient | |||||||||||
\(\ds \) | \(=\) | \(\ds \map f x \map {g'''} x + 3 \map {f'} x \map {g''} x + 3 \map {f''} x \map {g'} x + \map {f'''} x \map g x\) | Definition of Nth Derivative |
$\blacksquare$
Source of Name
This entry was named for Gottfried Wilhelm von Leibniz.
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 13$: Leibnitz's Rule for Higher Derivatives of Products: $13.48$