# Can the product rule be extended to the product of three functions?

Mar 22, 2017

We have:

$F \left(x\right) = f \left(x\right) g \left(x\right) h \left(x\right)$

Let

$u \left(x\right) = f \left(x\right) g \left(x\right) \implies F \left(x\right) = u \left(x\right) h \left(x\right)$

We can differentiate using the product rule:

$F ' \left(x\right) = u \left(x\right) h ' \left(x\right) + u ' \left(x\right) h \left(x\right)$ ..... [1]

And as $u \left(x\right) = f \left(x\right) g \left(x\right)$ we can similarly apply the product rule to get:

$u ' \left(x\right) = f \left(x\right) g ' \left(x\right) + f ' \left(x\right) g \left(x\right)$

Substituting this result, along with the definition of $u \left(x\right)$ into [1] we get:

$F ' \left(x\right) = f \left(x\right) g \left(x\right) h ' \left(x\right) + \left\{f \left(x\right) g ' \left(x\right) + f ' \left(x\right) g \left(x\right)\right\} \setminus h \left(x\right)$
$\text{ } = f \left(x\right) g \left(x\right) h ' \left(x\right) + f \left(x\right) g ' \left(x\right) h \left(x\right) + f ' \left(x\right) g \left(x\right) h \left(x\right)$

QED