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.
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Hey YDplusSB, the proof of the Product Rule that is presented on the wikipedia page is really cool! Thanks for mentioning that! I almost think it's too awesome to not describe on your blog in your own words... Is there anything about it that you find tricky? What makes the rectangle proof easier for you to understand than the other proofs they show on the wiki page? Can you see another way to divide the rectangle areas to prove the Product Rule in a different way?
ReplyDeletep.s. You're right to be careful with the information presented on wikipedia and it's good that you're cross-checking with your textbook!
Hi Cary,
ReplyDeleteI'm sorry that I cannot paste a picture here. But I think the tricky part in the proof by area is step 3. This is the step which you have to look at the rectangle to understand. I don't think the rectangle proof is easier in this particular case. Some people who learn better with images may benefit from that. I think it will still work if we cut the area like f(w)(g(w)-g(x)) + g(x)(f(w)-f(x)). By the way, thank you for commenting on my post.