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)
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Hey Marley, I'm your mentor. Your post is really clear and easy to understand, I love that you gave two different representations of the product rule.
ReplyDeleteI'm interested in your statement that "the book is not direct" since it proved the rule first. I don't have your textbook on hand, but are you sure it didn't state it before proving it? That sounds strange to me. It just launched into a proof of something without telling you what it was proving?
Anyway, I like your example, but it would be easier to read if you make it with the Equation Editor (its down in the righthand column). Its a bit tricky, but play around with it a bit, it makes things really pretty.
Great start!
Hello Kwad!
ReplyDeleteYes, the proof was done before the statement of the "product rule."
In my text book, the product rule is introduced by asking if the derivative of a product equals the product of the derivatives, just as it is for the sum (the derivative of a sum equals the sum of the derivatives). With an example after checking with nDeriv on the calculator, it was proven that this property doesn't work for multiplication.
So, they used the definition of derivative in order to find the product rule. This proof was a little long and confusing, and this is what made me lose my focus for a little bit.
Thank you for your comments,
next time I'll use the equation editor.
Marley.