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Re-Constructing the Quotient Rule

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


In other words, we need to know the derivative of the nested function


Do our chain rule stuff. The outside function is


and its derivative is


The inside function is


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.

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