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October 6th, 2012, 12:10 PM   #1
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2 questions (limits/derivatives)

Hello everyone. I have a few problems I am stumped on. Studying for a test on Monday and need some guidance. Here are the problems:

1) Limit as x approaches 0 of (5x) / tan(7x).

Can't plug in 0, as it will result in a 0 on the denominator. So I multiply by the reciprocal, which will be (5x) * [cos(x)/sin(x)].

And can not think of what to do next. Is it ok to plug the 0 into the denominator of the recriprocal, which would result in a 1 in the denominator, and leave us with 5x * cos(7x)?

Or is that not the correct way to go about it. I don't know why but I always get stuck on these trig limit problems.

2) Given that f(x) = 3xh(x), h(2) = -1, and hprime(2) = -1
Find fprime(2)

I know how to find derivatives I just don't know exactly how to go about this problem. Should I plug -1 in for x first?



In fact while I'm at it I might as well ask for suggestions on the best way to go about solving this:

find dy/dx of [(6x^2 - 1/x)]*(3x+1)

I am familiar with the product rule, quotient rule, and with how to find derivatives. I just know that there are sometimes easier ways than others to find them. For the problem above, I end up with a mass of jumbled numbers, letters, and exponents. I am hoping someone can tell me how the simplest way to do it.
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October 6th, 2012, 12:21 PM   #2
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Re: 2 questions (limits/derivatives)

1.)

Let



2.)



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October 6th, 2012, 12:47 PM   #3
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Re: 2 questions (limits/derivatives)

Quote:
Originally Posted by MarkFL
1.)

Let


^^^ Thanks. This is a method I don't believe anyone has ever shown me. Is there a more commonly used method? Just so I can know in case I get stuck on another problem. If I multiply by the reciprocal (which would be 5x * cos7x / sin(7x) ), I am just wondering what would be the next step. Really I guess what I'm asking is how would you get the derivative of cos(7x) / sin(7x). I guess here I could just plug in 0, which would be cox(x) / 1, but is that correct?

Quote:

2.)

<<< Not sure what you did here. What happened to the x in the 2nd term? Why is the x gone but the h still there?

<<< So just take the derivative and then plug in 2 for x. Got it.

[/quote]

Not sure if you noticed but I did edit my post so if anyone can help me there is a 3rd problem in my original post I am also stuck on.
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October 6th, 2012, 01:12 PM   #4
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Re: 2 questions (limits/derivatives)

If you know L'H˘pital's rule (most times it seems students haven't learned this yet, that's why I didn't use it), since you mention differentiation, then you could simply write:



For your third problem, I would write:



Now, using the product rule, we find:



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October 6th, 2012, 01:30 PM   #5
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Re: 2 questions (limits/derivatives)

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Originally Posted by MarkFL
If you know L'H˘pital's rule (most times it seems students haven't learned this yet, that's why I didn't use it), since you mention differentiation, then you could simply write:


^^^ Can you tell me how and why the denominator, tan(7x), which has a derivative of sec^2(7x) (that I do know), but I don't know why it is multiplied by a 7 (the 7 in front of sec^2(7x) ). Is that another derivative process that I am not seeing? I've been known to get lost doing problems because I could never figure out when and where the chain rule was supposed to be applied.

Quote:

For your third problem, I would write:



Now, using the product rule, we find:



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October 6th, 2012, 01:36 PM   #6
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Re: 2 questions (limits/derivatives)

Yes, the coefficient of 7 comes from the application of the chain rule as .
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October 6th, 2012, 01:40 PM   #7
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Re: 2 questions (limits/derivatives)

Quote:
Originally Posted by MarkFL
Yes, the coefficient of 7 comes from the application of the chain rule as .
Ok thanks. I feel more comfortable using that method for now.

Oh and I still am not quite sure about problem 2.

Not sure if you noticed I typed my message into the quote below:

Quote:

2.)

<<< Not sure what you did here. What happened to the x in the 2nd term? Why is the x gone but the h still there?

<<< So just take the derivative and then plug in 2 for x. Got it.

[/quote]

What process/rule was done to solve this problem? And also, as I noted above, in the 2nd part What happened to the x in the 2nd term? Why is the x gone but not the h?
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October 6th, 2012, 01:49 PM   #8
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Re: 2 questions (limits/derivatives)

I used the product rule:

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October 6th, 2012, 01:56 PM   #9
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Re: 2 questions (limits/derivatives)

Quote:
Originally Posted by MarkFL
I used the product rule:

Gotcha. I guess I am having trouble telling where one function stops, and the other begins.
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October 6th, 2012, 04:21 PM   #10
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Re: 2 questions (limits/derivatives)

.............

EDIT
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