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

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Answer 1 · 2017-03-22T20:57:13

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) # we can similarly apply the product rule to get:

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

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

# 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