Leibniz's Rule/One Variable/Second 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 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:

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

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 $\ds$ $=$ $\ds \binom 2 0 \map {f^{\paren 0} } x \map {g^{\paren 2} } x + \binom 2 1 \map {f^{\paren 1} } x \map {g^{\paren 1} } x + \binom 2 2 {f^{\paren 2} } x \map {g^{\paren 0} } x$ $\ds$ $=$ $\ds \map {f^{\paren 0} } x \map {g^{\paren 2 } } x + 2 \map {f^{\paren 1} } x \map {g^{\paren 1} } x + \map {f^{\paren 2} } x \map {g^{\paren 0} } x$ 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.