3.2 The Product and Quotient Rules

Greetings, comrades. Happy Tuesday! Since Tuesday is the worst day of the week (ask me in class if you need an explanation as to why this is true), I thought we could all use a little cheering up. Hey, let's learn some math to make us all feel better!

Today we learned Chapter Three, Section Two: The Product and Quotient Rules.

When asked to find the derivative of a product of two functions, our INSTINCT might be to multiply the derivatives of the two individual functions. In other words, if h(x) is equal to the product of g(x) and h(x), then the derivative of h(x) should be equal to the product of the derivative of g(x) and the derivative of f(x). It seems like that would be true...but no. WRONG! STOP! DO NOT DO THAT!

Let's look at why:


Okay, so we have established what IS NOT the product rule. I am sure you are all dying in anticipation and need to find out what the product rule IS before you spontaneously combust.

THE PRODUCT RULE



In words: h(x) represents the product of f(x) and g(x). To find the derivative of h(x) , you must first find two components: the first component is the product of f(x) and the derivative of g(x). The second component is the product of g(x) and the derivative of f(x). Then take the sum of these two components, and holy mackeral, you've found the derivative of h(x).

An example...


FYI -- You can leave it like this (as factors). On tests/quizzes, you do not have to multiply it all out into a big messy equation.

Okay, so now you guys are feeling all cool because you know the product rule. Well, I've got news for you, there is MORE! (I know you are cheering, even though you are sitting at your computer by yourself right now. You are just jumping up and down in your chair and yelling, "Yay! Yay! Yay!" If you aren't, there's nothing I can do, but you really should be. )

THE QUOTIENT RULE

I know what you are all thinking. Holy cow, that is a lot to process. But, NEVER FEAR, THE GODS OF RHYMING AND OTHER VERBAL NICETIES ARE HERE! Calculus folk have come up with a little song/rhyme/chant to help us remember the quotient rule. YAY!

"Lo dHi - Hi dLo over the square of what's down below."

For all of you visual learners:



And, finally, an example to illustrate:



If you want some extra practice, here is a link to a website about the Product and Quotient Rules (the home page of this link is "http://www.coolschool.ca," which I think is pretty, well, cool!)

http://www.coolschool.ca/lor/CALC12/unit3/U03L02.htm

Now, I have a grand surprise for everyone....the transcript of an EXCLUSIVE interview with none other than Mr. French, the cult god of Calculus AB!

Kristin: Okay, Mr. French, first question. When did you first discover your love of math?
Mr. French: There wasn’t really a single moment. I was good at math throughout high school. This was back in the time when there were no extra points for AP classes, or anything. And we had to walk uphill in the snow barefoot to get to school (it takes me a while to figure out he is kidding)…I think of math as a puzzle, and one way to solve problems. Math is not my passion; problem solving is.
Kristin: What is your favorite TV show?
Mr. French: Studio 60 on the Sunset Strip. If you liked West Wing, you will love Studio 60. (I happen to agree, Studio 60 is amazing! Everyone watch on Monday nights at 10:00.)
Kristin: What is your favorite thing about teaching, and your least favorite thing about teaching?
Ami (who is in the room for tutoring): Ryan.
Mr. French: Well, my favorite thing is working with students who want to learn, and watching the light bulb come on. My least favorite is definitely grading. It can be a little painful.
Kristin: Do you ever feel like you have failed if everyone gets bad grades on something?
Mr. French: Well, yeah, it can be hard. Either you have failed me, or I have failed you.
Kristin: What is your favorite holiday?
Mr. French: Christmas, because it is one of my few opportunities to spend time with my family.
Kristin: If you could be any character from a Disney movie, who would you be?
Mr. French: (after a lot of pondering, as there are so many superb Disney films to choose from, as we all know.) Woody, from Toy Story. You know, the whole Western thing.
Kristin: If you had to describe calculus in one word, what would it be?
Mr. French: Hmm…That sounds like my Stanford essay.
Kristin: What was the question?
Mr. French: You had to choose one word to describe yourself.
Kristin: And your answer was?
Mr. French: Renaissance man.
Kristin: Isn’t that two words?
Mr. French: Yes, I addressed that very issue in my first sentence. “Renaissance man” is an entry in the dictionary, so I used it as a single word.
Kristin: Anyway, one word for calculus?
Mr. French: Puzzling.
Kristin: Who is your favorite country music artist? (I remember from JPD that Mr. French is into country.)
Mr. French: It changes all the time. Right now, Terri Clark. I saw her at House of Blues a few months ago. She was really good at having a conversation with the audience. One time I saw Leann Rimes, who has a great voice, in concert, and she started talking about all of her deep, emotional life experiences and how they had affected her music, at the age of 21.
Kristin: What were your first impressions of Calculus AB Section B?
Mr. French: Energetic and inquisitive.
Kristin: Is that a good thing?
Mr. French: Well, yeah, it’s why I love to teach. But then there is also “a little hard to control…”
Kristin: Is that us?
Mr. French: Well, what do you think?
Kristin: Well.......anyway, last question! If you could give one piece of advice to high school students, what would it be?
Mr. French: Did you read the quote from the Chapter Two Test Topics blog posting?

(At first I feel bad that I have no idea what Mr. French was talking about, but when he puts the blog up on the projector, I realize that the quote was from another section's blog. Anyway, the quote is: “There is nothing noble in being superior to someone else. The true nobility is in being superior to your previous self.” – Hindu Proverb)

See everybody tomorrow....Lisa, you're next in line for the transcendental experience known as class blog posting.