Leibniz's Rule/One Variable/Third Derivative

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.