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.

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