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Handout Daily Prep 2.3 - The Product and Quotient Rules

Section Overview

This section covers the following concepts: The product and quotient rules for differentiation.

Section Basic learning objectives

These are the tasks you should be able to perform with reasonable fluency when you arrive at our next class meeting. Important new vocabulary words are indicated in italics.

Apply all the derivative rules from sections 2.1 and 2.2 with fluency.
Give a specific example to show that the derivative of a product of two functions, $f(x)\cdot g(x)\text{,}$ is not the product of the derivatives, $f'(x)\cdot g'(x)\text{.}$
State and apply the product rule.
State and the quotient rule.

Section To prepare for class

Complete all actions listed below. Respond to the questions highlighted with Submit.

Read motivating questions and the introduction to section 2.3 (up until Preview Activity 2.3.1).
📝 [Submit] Do Preview Activity 2.3.1 (via hand calculations).
📝 [Submit] Check your answers to Preview Activity 2.3.1 using GeoGebra. Submit a screen capture of your work along with responses to the questions posed.
- Enter the function $f(t) = 2t^{2}$ by typing 2*t 2 into the first input box. GeoGebra will define and label this function as $f(t)$ for you.
- Enter the function $g(t) = t^{3}+4t$ by typing t 3 + 4*t into the second input box. GeoGebra will define and label this function as $g(t)$ for you.
- In the third input box, type f(t)*g(t). Then, relabel h as p (if you wish...just like in the activity). [To relabel, just use backspace.]
- (Optional) If you like, you can simplify $p(t)\text{.}$ In this next input box, just type Simplify(p(t)). Nice, yes?
- GeoGebra has the ability to return derivatives of functions. In the next input box, type Derivative(p(t)). GeoGebra will automatically label the result as $p'(t)\text{.}$ Does this match your hand-calculated result?
- In the next input box, we calculate $f'(t)\cdot g'(t)$ by typing Derivative(f(t))*Derivative(g(t)). Is the result the same as $p'(t)\text{?}$ [If needed, the Simplify function (as described above) can be used.
- Do parts (d) and (e) of Preview Activity 2.3.1 using GeoGebra to do the calculations.
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Read section 2.3.2.
Watch video Product Rule Examples (6:47).
Prompt Copilot “I don’t want a proof, but please explain in simple terms how the product rule in calculus comes about.”
Prompt Copilot “Create an example of a typical product rule question in a college-level calculus class using a numerical table. Be sure to provide me the solution too.” Is the solution provided correct?
Read section 2.3.3.
Watch video Quick Review: The Product and Quotient Rules (1:59).
Watch video Quotient Rule Examples (10:37).
Do Exercise 1 in section 2.3.6. Submit a screenshot to demonstrate completion.
Do Exercises 2-10 in section 2.3.6.

Checkpoint 57. 📝 [Submit] Which Rule Do You Need?

Sort each function according to the primary differentiation rule that is required. Do not differentiate the function. Focus only on recognizing its structure.

$x^2\sin(x)$
$(x^3+1)e^x$
Product Rule
$\dfrac{\sin(x)}{x}$
$\dfrac{x^2+1}{e^x}$
Quotient Rule
$5x^4-2x+7$
$\cos(x)+x^3$
No Product or Quotient Rule Needed

Section After class

Solidifying the concepts discussed in class through practice is necessary to build your skills.

Read section 2.3.4.
Watch video Combining the Product and Quotient rules (9:00).
Do these problems.
1. Suppose the $f$ and $h$ are functions and that $f(3)=2\text{,}$ $f'(3) = -2\text{,}$ $h(3) = 1\text{,}$ and $h'(3) = 4\text{.}$
  1. Calculate $m'(3)\text{,}$ where $m(x) = f(x) \cdot h(x)\text{.}$
  2. Calculate $p'(3)\text{,}$ where $\displaystyle p(x) = \frac{f(x)}{x^{2}h(x)}\text{.}$
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2. Use information from the graphs of $f(x)$ and $g(x)$ and the tangent lines shown in Figure 58 to calculate each derivative.
  
  $Two separate coordinate graphs are shown. On the left, a smooth blue curve labeled f(x) increases from left to right. Two filled points on the curve are labeled with coordinates (2, 3) and (4, 4.8).$
  
  $On the right, a smooth blue curve labeled g(x) decreases from left to right. A horizontal segment at y equals 5 includes two filled endpoints labeled (0, 5) and (2, 5). Farther to the right, a filled point on the curve is labeled (3.5, 3), with a straight black line tangent to the curve at that point.$
  
  Figure 58. The graph of $f(x)$ and $g(x)$ and two particular tangent lines.
  1. $h'(2)$ if $h(x) = x^{2} f(x)$
  2. $k'(2)$ if $\displaystyle k(x) = \frac{f(x)}{g(x)}$
  3. $m'(2)$ if $\displaystyle m(x) = \frac{x g(x)}{f(x)}$
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Do Exercises 11-12, 14 in section 2.3.6.
Read section 2.3.5 (the summary).
Do Exercises 13,15 in section 2.3.6.
Start working on the MOMwork (MyOpenMath) assignment for this section.

Section Advanced learning objectives

In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

Prove the quotient rule using the product rule.
Differentiate a function for which the derivative involves a combination of the product rule, quotient rule, and other differentiation rules.
Use the product and quotient rules in the context of a real-world problem to find the slope of a tangent line, the instantaneous rate of change in a function, or the instantaneous velocity of an object.

Section Additional suggestions

Watch this introductory video to help in your reading of section 2.3.1.
Watch video computing derivatives using graphical information (6:49).

Section Answers

1. $m'(x)=f'(x) \cdot h(x) + f(x) \cdot h'(x)\text{.}$ So, $m'(3) = f'(3) \cdot h(3) + f(3) \cdot h'(3) = (-2)(1) + (2)(4) = 6$
2. $\displaystyle \displaystyle p'(3)=-\frac{82}{27}$
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1. $\displaystyle h'(2)=15.6$
2. $\displaystyle k'(2)=0.18$
3. $\displaystyle \displaystyle m'(2)=\frac{2}{3}$
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