Question

How does the quotient rule differ from the product rule?

Answer 1

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)`