Leibniz's Rule/One Variable/Second 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 twice differentiable.


Then:

$\paren {\map f x \map g x} = \map f x \map {g} x + 2 \map {f'} x \map {g'} x + \map {f} x \map g x$


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 $n = 2$:

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

$\blacksquare$


Source of Name

This entry was named for Gottfried Wilhelm von Leibniz.


Sources

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