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

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

Setting $n = 2$: