How does the quotient rule differ from the product rule?

1 Answer
Mar 29, 2015

The simple answer is that the
Quotient Rule:
#(d(g(x)/(h(x))))/(dx) = (g'(x)h(x)-g(x)h'(x))/(h(x))^2#
is used when you want to find the derivative of a function which can
easily be viewed as one function divided by another function.
For example
#f(x) = (3x^2+7)/(sqrt(x))#
can be viewed as
#f(x) = (g(x))/(h(x))#
with #g(x) = 3x^2+7# and #h(x)=sqrt(x)#

The Product Rule:
#(d(g(x))*(h(x)))/(dx) = g'(x)*h(x)+ g(x)*h'(x)#
is used when you want to find the derivative of a function which can be easily viewed as one function multiplied by another function.
For example
#f(x) = (3x^2+7)*sqrt(x)#

The more complex answer is that they aren't really different rules:
Given a function
#f(x) = (g(x))/(r(x))#
if you replace #r(x)# with #1/(h(x)# or #h(x)^(-1)#
and apply the Product Rule
(with some care)
you will end up with the Quotient Rule.
[if you decide to try this you will need to remember #(d(h(x)^(-1)))/(dx) = (-1)(h(x)^(-2))*((d h(x))/dx)#