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

##### 1 Answer

Mar 22, 2017

We have:

# F(x)=f(x)g(x)h(x) #

Let

# u(x)=f(x)g(x) => F(x) = u(x)h(x) #

We can differentiate using the product rule:

# F'(x) = u(x)h'(x) + u'(x)h(x) # ..... [1]

And as

# u'(x)=f(x)g'(x) + f'(x)g(x) #

Substituting this result, along with the definition of

# F'(x) = f(x)g(x)h'(x) + {f(x)g'(x) + f'(x)g(x)} \ h(x) #

# " " = f(x)g(x)h'(x) + f(x)g'(x)h(x) + f'(x)g(x)h(x) #

QED