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:
 * $\displaystyle \left({f \left({x}\right) g \left({x}\right)}\right) = f \left({x}\right) g \left({x}\right) + 2 f' \left({x}\right) g' \left({x}\right) + f'' \left({x}\right) g \left({x}\right)$

where $\left({n}\right)$ denotes the order of the derivative.

Proof
From Leibniz's Rule:
 * $\displaystyle \left({f \left({x}\right) g \left({x}\right)}\right)^{\left({n}\right)} = \sum_{k \mathop = 0}^n \binom n k f^{\left({k}\right)} \left({x}\right) g^{\left({n - k}\right)} \left({x}\right)$

Setting $n = 2$: