Product Rule

I thought that the textbook gave a good explanation of the product rule. The examples of the product rule being done on functions was easy to follow and it had a box stating the product rule verbally. "Derivative of first times second, plus first times derivative of second..And don't forget the chain rule". Although the textbook was good, the site I found was really nice because it had figures that moved and showed the product rule in affect. The figure showed the relation the two functions and their derivatives and you could manipulate the components. Overall, I liked the site better because of the figure but if I had not read the textbook first, I may have not understood the site as much. The product rule as the site stated, is {f(x)g(x)}^'=f(x)g^'(x) +f^'(x)g(x).

ex.  y = x²(3x+1)

First you take the derivative of the first equation (2x) multiplied by the second, which equals 6x^2+2x

Plus the derivative of the second equation (3) multiplied by the first, which is 3x^2

and it all comes out to 9x^2+2x.....I think

Let me know if this is right!

Product Rule Scribe Post

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Live from Carbondale, it's Tuesday night! Welcome to your first scribe post. Our first scribe post is about the product rule, and since I can't think of any cool pictures that would represent the product rule, I'm just going to jump right in.

We started exploration 4-2 in class on Monday with this objective: Given a function that is a product of two other functions, find in one step, an equation for the derivative function. In math terms: if f(x)=g(x)*h(x), what will f'(x) be? First, we figured out if differentiation distributes over addition through this test:

• Using a function that was a sum of two functions,:

  y=(x³)+(5x+1)

• we first differentiated by adding the derivative of each separate function:

  d/dx(x³)+d/dx(5x+1)

• this equalled:

  =(3x²)+(5)

• Then we differentiated the function as a whole:

  d/dx(x³+5x+1)

• this equalled:

  =(3x²+5)

As you can see, the derivative of a sum of two funtions is equal to the sum of the derivatives of the two functions. This is how we discovered that YES, differentiation does distribute over addition. The we wondered, does it distribute over multiplication? We used similar steps to answer this question:

• we started with the same composition of equations, but this time they were multiplied:

  y=(x³)*(5x+1)

• we differentiated each equation seperately:

  d/dx(x³)*d/dx(5x+1)

• this equalled

  (3x²)*(5)

• which simplifies to:

  15x²

• then we wanted to differentiate as a whole equation, so first, we simplified by distributing the x³ to the 5x+1 which equalled:

  5x⁴+x³

• then we differentiated the equation as a whole:

  d/dx(5x⁴+x³)

• This equals:

  20x³+3x²

As you can see from this example, the derivative of a product of two functions is not equal to the the product of the derivatives of the two functions. This is how we discovered NO differentiation does NOT distribute over multiplications. So then we knew that we needed to find a way to find the derivative of a product of two functions. We then used the definition of derivative to derive the formula for the derivative of a product of two functions. Below is a slide from Bru's presentation that has the answers to the proof on the second page of your exploration. Below is an explanation of each step.

I'm really sorry if you can't read this, the same thing can be found of the slide show if the product rule that Bru posted.

1.-2.: Since y=uv, Δy=ΔuΔv and

(y+Δy)= (u+Δu)(v+Δv)

2.-3.: FOIL

3.-4.: The positive uv at the beginning and the negative uv at the end add to 0.

3.-4.: explained

4.-5.: You want the Δu or Δv to be in the fraction and the u or v to next to it. Bru did most of this step for you.

5.-6.: Explained, take the limit of all three seperately.

6.-7.: Δ symbol changes to d to symbolize that the limit is a derivative. The Δu in the third becomes zero based on the graph at the top of the page on the far left. We are taking the limit as Δx approaches 0 and as you can see by this graph, as Δx approaches 0, Δu also approaches 0.

So, the product rule is:

If f(x)=g(x)h(x)

Then f'(x)=g'(x)h(x)+g(x)h'(x)

We ended class with a couple of practice problems and learned that the algebra part (simplifying) is actually the hardest part. Here is one example we did in class. My commentary and additions are in red.

Check Your Understanding:

2. If f(x)=[(3x-8)⁷][(4x+9)⁵], find f'(x)

   I put in the red brackets to help me figure out what the two different functions are.

f'(x)= (7(3x-8)⁶•3)((4x+9)⁵)+((3x-8)⁷)•(5(4x+9)⁴•4)

Take a look at the first term. In this term, you differentiated the function (which happened to be a composite functions and required the chain rule), and multiplied it by the second function. Then in the second term, you took the first function and multiplied it by the derivative of the second function (which was also composite). As long as you can keep all this straight in your head, this step is relatively easy. But now you have to simplify.

f'(x)= (3x-8)⁶(4x-9)⁴[21(4x+9)+20(3x-8)]

In this step you factored out (3x-8)⁶ and (4x-9)⁴. Then, you were only left with what is inside the brackets.

f'(x)= (3x-8)⁶(4x-9)⁴[84x+189+60x-160]

Distribute within the brackets.

f'(x)= (3x-8)⁶(4x-9)⁴[144x+29]

Combine like terms within the brackets, and now the equation is fully simplified.

My answers for numbers 3, 4, and 5. Let me know if you agree or disagree.

3. e^sinxcos²x-e^sinxsinx

4. x^-6.3(1/x+ln4x*-6.3x^-1.3)

5. (21x⁶-60x4)cos10x-10(3x⁷-12x⁵)sin10x

   I think this one can be simplified more. I tried to factor out 3x⁴ but it didn't work, or maybe I did it wrong.

Enjoy! Comment if you wish.

Next Scribe is Babar.

Product Rule

I Google searched the product rule and I clicked on the second search result, (the first was Wikipedia). It was a fairly dull looking website, but it was simple and to the point. First it stated the definition of the product rule, and was then followed by about 15 problems to work on. Each problem also had a link to a detailed solution, so I could easily check my work, or have something to go to if I got confused.

Here's the link to the site that I found:
http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/productruledirectory/ProductRule.html

I think in this case the online source was more valuable to me, because it gave me the exact information I needed as well as an opportunity to try out that knowledge. The book does the same thing, but it also has a lot more information to navigate around.

One other thing I think is helpful, is being able to verbally define the Product Rule. This was not included in either the website or the book, but it can be done without them.

Here is Definition of the Product Rule:


Example:
Differentiate .

Solution:

Today's Slides: January 11

Hello All,
Here are the slides from today's class on the Product Rule....

CRMS Calculus 2010 January 13, 2010

View more presentations from Colorado Rocky Mountain School.

Cheers, Bru

The Product Rule

So the online source that I found was from HMC and was short, succinct, and easy to read. The examples were really easy for me to understand and the explanations of the Product Rule made sense. I like this better than the textbook which seemed to me to be longer than it needed to be. The "fairly complicated algebra" passed me by and I found that the Property section was all I needed to understand the rule. I definitely feel that the online source was more useful to me.
If I had wanted all the long algebra they gave the link, just in case.

This definition and example come from

http://www.math.hmc.edu/calculus/tutorials/prodrule/

Definition:

h(x)=f(x)g(x)=f(x)g(x)+f(x)g(x)

Example:

If h(x)=x2sinx then

h(x) = =2xsinx+x2cosx = (x2)sinx+(x2)(sinx)

Product Rule

I am a visual learner so the video was definitely more helpful for me. I liked the video from online because it showed exactly how to use the product rule on a function. The demonstration included different colors, sound, and also a clear visual representation of how to use the product rule. The book gave a clear and helpful description as well however personally I just have a hard time concentrating on something that is printed in a thick text book. An upside to the book was that there was more than one example.

Definition:

If f(x)=g(x)h(x) then f'(x)=g'(x)h(x) + g(x)h'(x)

Example:

f(x)=(4x^3)(3x^6)

f'(x)=(12x^2)(3x^6)+(4x^3)(18x^5)

f'(x)=(36x^8)+(72^8)

f'(x)=108^8

Example:
f(x)=(4x^3)ln5x

f'(x)=(12x^2)(ln5x)+(4x^3)((1/5x)(5))

f'(x)=(12x^2)(ln5x)+(4x^3)(1/x)

Product Rule Site Comparison

Today I just googled 'product rule' and looked at one of the options that came up. From just looking at one site, I understood the product rule. The site I looked at: first gave a verbal definition and also an algebraic definition. Then, it had a whole list of problems that you could look at a detailed solution to. It included all kinds of problems full of exponents, trig, ln, etc. Also, in the solutions, each step was clearly written out and it was easy to see what had been done. When I looked at the book, however, I was confused, even though I had just learned the product rule. The verbal descriptions it had were very complex and confusing and I could not follow them. One thing I didn't like about the web site I found, though, was that there was not a verbal description for each of the steps in the solutions, which the text book offers. It helps me to see example with real numbers instead of just variables, but it also helps me to read how each step was done.
Definition:
if h(x)=f(x)g(x)
then h'(x)=f'(x)g(x)+f(x)g'(x)

Example:
if: h(x)= (x^2+5)(3x)
then: h'(x)= (2x)(3x)+(x^2+5)(3)
(derivative of the first)(the second)+(the first)(derivative of the second)
=3x^2+6x^2+15
=9x^2+15

Product Rule

While both explanations where helpful in explaining the Product Rule the internet site I found would not have been enough with the book. The book was extremely clear in its definition while also providing a straight forward expample to begin creating further understanding with. As well as the verbal definition, the derivative of the first times the second, plus the first times the derivative of the second. However the site provide a number of more complex examples which could prove to be more useful in the future and more applicable during further practice. Unfortunately the site was limited in the number of examples and was more focused on why this rule makes sense and how to prove that it actually works and is accurate. Ultimately both are moderately useful and work well together but would require more information for a successful watch it, do it, teach it, learning process.

http://en.wikipedia.org/wiki/Product_rule

y'=u'v+uv'
y=x^4cos6x
y'=4x^3cos6x+x^4(-sin6x)*6
y'=4x^3cos6x-6x^4sin6x

About Product Rule

I think the both of them help me understand the product rule. There is a lot of information online. There are some different proofs of product rule online. However, there is a lot of extra information online which I don’t really need. Sometimes we probably just want what we exactly need. Extra information would just make us confused. And a lot of the websites I found don’t provide me the right information. Internet is a really public place. Everyone can put up his or her own ideas online. So wrong information and fake information is inevitable. I think Wikipedia is a good place to find what we want. Even though everyone can edit the content of Wikipedia, most of the information appearing in Wikipedia is very useful and organized. We can simply find exactly what we need at the first sentence, and if we need more details, there is always extra information for us to look at. Textbook is also a useful source, and I feel better to read something that I am actually holding. It is also very simple to find the most important part of product rule in the textbook. The textbook highlights the definition and property of product rule and it is very easy to understand. Example questions and solutions help understand, too. I don’t really have a preference. I think both of the learning sources work for me really well.
Product rule:
Property from the textbook
If y = uv, where u and v are differentiable functions of x, then y' = u'v + uv'
Verbally: Derivative of first times second, plus first times derivative of second.

Simply, take the derivative of the first function times the second function, add this to the product of the derivative of the second fucntion and the first function.

Example:
y = (x^2)*(ln x)
y' = 2xln x + (x^2)*(1/ x) = x + 2xln x

http://en.wikipedia.org/wiki/Product_rule
Check out this.
The product rule is proved by the area of rectangles.
This is very neat.

Product Rule for visual Learners

When searching the internet for the product rule i stumbled upon a Youtube video. As a visual learner, i found it very helpful because the man gave an easy (non mathy) way to understand the procedure of the power rule. Also it allowed me to watch and listen to someone solving equations. However, the textbook gave a much more in depth explanation of what the Product Rule is. I found that the textbook was better for understanding the concept, but the video was much more helpful for explanations.

Product rule:

If f and g are differentiable functions, then the derivative of the product fg is:

(fg) '(x) = f(x) g '(x) + g(x) f '(x)

Example:f(x)=g(x)(h(x))f(x)=5x^3sin4xf'(x)=(15x^2 (sin4x))+(4cos(4x)(5x^3))

link:http://www.youtube.com/watch?v=uPCjqfT0Ixg

Product Rule

As in the process of learning, we should be able to "watch", "do", and "teach". The online definition was easier for me to "watch" because it had both visual and audio non complicated explanations. This helped me to understand what is the "product rule." On the other hand, the book's definition increased my understanding of the product rule because it has different examples. Therefore, if i was to chose between the text book and the internet, I would rather check on the internet first, since it's easier at the first glance, and then follow the text book later. If I didn't do it this way, I would certainly be confused by the text book, and it would take me longer to understand because the book is not direct - it proved the product rule, and then stated it later.

The product rule:

If f(x) = g(x) . h(x),

then f'(x) = g'(x) . h(x) + g(x) . h'(x)

In words: The derivative of a product of two functions equals the derivative of the first function times the second function, plus the derivative of the second function times the first function.

Example:

f(x) = 3x.cos(5x)

f’(x) = 3cos(5x) + 3x . – sin(5x).5

f’(x) = 3cos(5x) – 15xsin(5x)

Product Rule

Book vs. Internet

Looking in the book, I found that the definition was very concise and easy to understand. Unlike the site that I found which took time to watch and absorb all of the information as it flashed by on an animation. In the Presentation that was dragged on for several minutes I learned the same amount as in the two line definition box in the book. It is also a lot easier to understand an elementary explanation than a huge production with lots of distractions.

Definition:
if f(x) = u•v then f'(x) = u'•v + u•v'

Examples:
f(x) = 3x^2 • lnx
f'(x) = (6x • lnx) + (3x^2 • 1/x)
f'(x) = (6x • lnx) + (3x)

Awesome Product Rule Video!

I found this video to be exceptionally helpful. While the book did in fact strip everything of "frills," as previously stated, so did this video. It also provided examples, step by step, of different styles of a product property problem. The video used actual number examples, beyond "u" and "v", like the book does, which I find to be tremendously confusing. I'm hoping the source of this video, will be able to provide help with many more properties and rules.

Product Rule: Derivative of first times second, plus first times derivative of second

f '(x)= f '(x) • g(x) + f(x) • g'(x)

Example: h(x)= (sinx)(x^6 + x)

h '(x)= (cosx)(x^6 + x) + (sinx)(5x^5 + 1)

Product Rule

Personally, I found the website to be much better source for learning the product rule quickly and efficiently without compromising understanding. The book had much more lengthy and complete explanation of the product rule including graphs, verbal definitions, and symbolic definitions. Despite the thoroughness of the text book definition I still found the explanation on the website (http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/productruledirectory/ProductRule.html) to be much more useful for a person like myself. The online resource gave one short definition, but many full length examples of increasing complexity. I found this to be more useful than the textbook for mainly that very reason. It is often easier for me to understand a new concept by looking at multiple examples that I can easily refer to if I find myself stuck on a problem. Another additional reason as to why I prefer the online resource is because it is simpler and makes the product rule in calculus look less daunting while maintaining the information necessary. Hence, the online resource, oddly enough, gives me somewhat more confidence in my understanding of the concept.

Product Rule:
The product rule is a formal rule for differentiating problems where one function is multiplied by another. The rule follows from the limit definition of derivative and is given by where D stand for the derivative of a function (excerpted from http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/productruledirectory/ProductRule.html)

Example: Differentiate

Product Rule Source Response

I found the text book to be more helpful because it had less 'fluff' and was more concise. I feel that a good source on the internet could be as helpful if not more so than the text. However, the dealbreaker for me is finding that good source on the internet. It could take investigating up to 5 different sites in depth before I find a solid source. And if it's late on a school night and I have the option of opening the textbook for an easy explanation or digging through the internet hoping to find a good explanation, I choose option a. In the text, the concept is clearly put in a blue box so I know where the meat of the subject is, followed clearly by understandable examples. On the internet, who knows what I will find.
Thus, I choose the textbook because it is more reliable.

Product rule:

If y=uv
then
y'=u'v + uv'

Example of the product rule:

If y=5xlnx
then
y'=5lnx+5x(1/x)=5lnx+x

Product Rule

Or textbook clearly explained the definition of the derivative of a product of two functions. It also gave three examples in order to fully understand the product rule. The book includes both a mathematical and verbal explanation. The online explanation was similar in that it mathematically explained the product rule and gave multiple examples. I found the online explanation useful in that it explained the product rule in terms of f(x), g(x), and h(x), mathematically in multiple ways. I found the textbook explanation a bit easier to understand and visualize because of its verbal explanation.

Product Rule: Mathematical: If h(x)=f(x)g(x), then h'(x)= f'(x)g(x)+f(x)g'(x),

Verbally: The derivative of the function of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second.

Example:

f(x)=x⁴cos6x

f’(x)=4x³cos6x+x⁴(-sin6x)(6)

f’(x)=4x³cos6x-6x⁴sin6x

Product Rule

This forum post is showing the differences between textbook definitions and online definitions of the product rule. On our textbook, product rule was clearly defined with in the box. It also gave us the proof with the general definition of the derivative. It also provides simpler definition of the product rule with not only with letters, but also with words. On the other hand, online version of the definition of the product rule I found provided audio and video description of the definition. It was easier for me to understand with someone talking to me; therefore, I thought that online version I found had more clear description. However if someone wants to know more about product rule not simply about how to solve the problem, one should go through the textbook to learn about the product rule.

Product Rule:

Example:

Forum Question: The Product Rule

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The next derivative rule you will learn is the Product Rule for finding the derivative of a product of two functions. Search the Internet for a site which explains the Product Rule well enough for you to understand and apply. Bookmark this site on Delicious. So that your classmates can easily find this bookmark, you want to assign a tag which will be recognized by our class. The tag you will use for this bookmark (and all future bookmarks for this course) is crmscalc2010. Add a second tag of your nom de plume, and a third tag of Product Rule. Now the bookmark can be easily shared with anyone in our class. Look for your bookmark in our Delicious library on the left sidebar of our blog.

Once you have bookmarked the site, read Section 4.2 of our textbook on the Product Rule. Write a blog post comparing the explanations given on your bookmarked site and in the textbook. From which source did you get a better understanding of the rule? Include in your post what you liked and disliked about each explanation. End your post by stating the Product Rule and giving an example.

After posting to our blog, take this short quiz to check your understanding of the Product Rule.