One of the nifty tricks we can do with the chain rule is reconstruct the quotient rule. To find the derivative of
first rewrite the function like this:
h(x) = f(x) · (g(x))-1.
Now that h is written as a product, we can use the product rule to find its derivative:
h ' (x) = f ' (x)(g(x))-1 + f(x) · ((g(x))-1)'.
Now there are two big steps left. First we'll use the chain rule to find this derivative:
h ' (x) = f ' (x)(g(x))-1 + f(x) · ((g(x))-1)'.
Then we'll simplify the formula we got using the product rule until it magically turns into the quotient rule.
Chain Rule:
To finish applying the product rule, we need to know
((g(x))-1)'
In other words, we need to know the derivative of the nested function
(g(x))-1
Do our chain rule stuff. The outside function is
(□)-1,
and its derivative is
-(□)-2.
The inside function is
g(x),
and its derivative is
g ' (x).
Now we can use the chain rule:
((g(x))-1)' = -(g(x))-2 · g ' (x)
Since (g(x))-2 is the same thing as 
, we can rewrite this as
Returning to the product rule,
We can do some great simplifying here. Since (g(x))-1 is the same thing as 
, we can rewrite this as
Now we can put the fractions over a common denominator and combine them:
Ta-daa! This is the quotient rule.