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

Simplifying:

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.