Section 2.4 The product and quotient rules (DF4)
Learning Outcomes
Compute derivatives using the Product and Quotient Rules.
Activity 2.4.1.
Let
(a)
Find
(b)
Let
(c)
True or false:
Theorem 2.4.2. Product Rule.
If
Activity 2.4.3.
The product rule is a powerful tool, but sometimes it isnβt necessary; a more elementary rule may suffice. For which of the following functions can you find the derivative without using the product rule? Select all that apply.
Activity 2.4.4.
Find the derivative of the following functions using the product rule.
(a)
(b)
(c)
Activity 2.4.5.
Let
(a)
Determine
(b)
Let
(c)
True or false:
Theorem 2.4.6. Quotient Rule.
If
Activity 2.4.7.
Just like with the product rule, there are times when we can find the derivative of a quotient using elementary rules rather than the quotient rule. For which of the following functions can you find the derivative without using the quotient rule? Select all that apply.
Activity 2.4.8.
Find the derivative of the following functions using the quotient rule (or, if applicable, an elementary rule).
(a)
(b)
(c)
(d)
Activity 2.4.9.
Demonstrate and explain how to find the derivative of the following functions. Be sure to explicitly denote which derivative rules (product, quotient, sum and difference, etc.) you are using in your work.
(a)
(b)
(c)
Note 2.4.10.
We have found the derivatives of
Activity 2.4.11.
Consider the function
(a)
What is the domain of
(b)
Use the quotient rule to show that one expression for
(c)
Which trig identity might be useful here to simplify this expression? How can this identity be used to find a simpler form for
(d)
Recall that
(e)
For what values of
Activity 2.4.12.
Let
(a)
What is the domain of
(b)
Use the quotient rule to develop a formula for
(c)
Use other relationships among trigonometric functions to write
(d)
What is the domain of
Activity 2.4.13.
Let
(a)
What is the domain of
(b)
Use the quotient rule to develop a formula for
(c)
Use other relationships among trigonometric functions to write
(d)
What is the domain of
Activity 2.4.14.
Let
(a)
What is the domain of
(b)
Use the quotient rule to develop a formula for
(c)
Use other relationships among trigonometric functions to write
(d)
What is the domain of
Theorem 2.4.15.
We can now summarize the derivatives of all six trigonometric functions.
Activity 2.4.16.
Consider the functions
and the function
In answering the following questions, be sure to explicitly denote which derivative rules (product, quotient, sum/difference, etc.) you are using in your work.
(a)
Find the derivative of
(b)
Find the derivative of
(c)
Find the value of the derivative of
(d)
Find the value of the derivative of
(e)
Consider the function
Find
Activity 2.4.17.
(a)
Differentiate
(b)
What do you expect the derivative of
(c)
What do your answers above tell you above the shape of the graph of
Activity 2.4.18.
The quantity
(a)
What does the data provided tell you about the sales of skateboards?
(b)
The total revenue,
(c)
Find the derivative of the revenue when
(d)
What is the sign of the quantity above? What do you think would happen to the revenue if the price was changed from $140 to $141?
Activity 2.4.19.
Let
(a)
Let
(b)
Let
(c)
How would you explain the practical meaning of your findings to a driver who knows no calculus?
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