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:

$\displaystyle \paren {\map f x \, \map g x}^{\paren n} = \sum_{k \mathop = 0}^n \binom n k \map {f^{\paren k} } x \, \map {g^{\paren {n - k} } } x$

where $\paren n$ denotes the order of the derivative.


Setting $n = 3$:

\(\displaystyle \paren {\map f x \, \map g x}'''\) \(=\) \(\displaystyle \paren {\map f x \, \map g x}^{\paren 3}\) Definition of Nth Derivative
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 0}^3 \binom n k \map {f^{\paren k} } x \, \map {g^{\paren {n - k} } } x\) Leibniz's Rule
\(\displaystyle \) \(=\) \(\displaystyle \binom 3 0 \map {f^{\paren 0} } x \, \map {g^{\paren 3} } x + \binom 3 1 \map {f^{\paren 1} } x \, \map {g^{\paren 2} } x + \binom 3 2 \map {f^{\paren 2} } x \map {g^{\paren 1} } x + \binom 3 3 \map {f^{\paren 3} } x \, \map {g^{\paren 0} } x\)
\(\displaystyle \) \(=\) \(\displaystyle \map {f^{\paren 0} } x \, \map {g^{\paren 3} } x + 3 \map {f^{\paren 1} } x \, \map {g^{\paren 2} } x + 3 \map {f^{\paren 2} } x \, \map {g^{\paren 1} } x + \map {f^{\paren 3} } x \, \map {g^{\paren 0} } x\) Definition of Binomial Coefficient
\(\displaystyle \) \(=\) \(\displaystyle \map f x \, \map {g'''} x + 3 \map {f'} x \, \map {g''} x + 3 \map {f''} x \, \map {g'} x + \map {f'''} x \, \map g x\) Definition of Nth Derivative

$\blacksquare$


Source of Name

This entry was named for Gottfried Wilhelm von Leibniz.


Sources