# 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

((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.
The passage describes the process of deriving the quotient rule from the product and chain rules in calculus. It does not contain any specific mathematical questions or answers to such questions.
no_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# 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

((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.
user
The passage describes the process of deriving the quotient rule from the product and chain rules in calculus. It does not contain any specific mathematical questions or answers to such questions.

TYPE: NO_QUESTION