Calculus Derivatives, OMG!!!

This is a discussion on Calculus Derivatives, OMG!!! within the A Brief History of Cprogramming.com forums, part of the Community Boards category; That's it! I'm so damn frusterated I... I.. AHHH!! So, it's 2:07am and I work in afew hours. I just ...

  1. #1
    Xei
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    Calculus Derivatives, OMG!!!

    That's it! I'm so damn frusterated I... I.. AHHH!!

    So, it's 2:07am and I work in afew hours. I just finished my derivatives exam through online school and I got 100%. But that's not the point, the point is that there are still some questions that are ........ing me off. For instance:

    The book says:

    "Find the second derivative of: ((x^2)+1)^(1/2)"

    Okay, "Simple", I think. I found the first derivative to be:

    x((x^2)+1)^(-1/2)

    The book agrees with that answer, except for the second derivative. The book believes that the second derivative is ((x^2)+1)^(-3/2) which makes no sense at all. How is this possible? I tried using the Product Rule w/ Chain Rule which would normally go like this:

    X = V
    ((x^2)+1)^(-1/2) = U

    V ' = Derivative of V
    U ' = Derivative of U

    So now, my derivative should be:

    (V ')(U) + (U ')(V) = 0

    Which is quite long for me to type out, but anyways, I end up with an answer which should include X^2, but it doesn't. I'm frusterated, this makes no sense. Please help.
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    The Earth is not flat. Clyde's Avatar
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    Hmmm i'm also not getting the book's answer, maybe its wrong?

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    Registered User axon's Avatar
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    The book is correct my friend...I'm really short of time right now, so I will post the proper way to derive f''(x) when I get back in front of the computer.

    But I have done the problem and the book is indeed right.

    some entropy with that sink? entropysink.com

    there are two cardinal sins from which all others spring: Impatience and Laziness. - franz kafka

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    Registered User axon's Avatar
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    Well, I did find some time to type it out, here it is:
    Code:
    we agree that 
    
    	f'(x) = x / (x^2+1)^(1/2)
    
    Now, to compute f''(x) you can bring the denominator up or leave it there.
    If you leave it, the derivation will be a bit harder, but simplification 
    much easier.
    
    	f''(x) =[ (x^2+1)^(1/2) - x(x^2+1)^(-1/2) ] / (x^2 + 1) 
    	//to simplify brek into two fractions
    	       =[(x^2+1)^(1/2) / (x^2+1) ] - [ x / {(x^2+1)^(1/2)(x^2+1)}
            //from now on its just algebra, take common denoms
                    = (x^2 + 1 - x^2) /  [(x^2+1)^(1/2)(x^2+1)]
    	//x^2 in the numerator cancel out and the denom simplifies to...
    	       = 1 / (x^2+1)^(3/2)
    often times the hard part of calculus is the agebra...as weird as it sounds. I know that my professors on exams accepted not simplified solutions. If you have a TI calculator, you could put in your solution and equate it with the book's, and see if you get a 'true'. That way you'll know if you have the same answer just unsimplified.

    axon

    EDIT:: write what i 'coded' on paper; it will make more sense
    Last edited by axon; 10-23-2003 at 06:56 AM.

    some entropy with that sink? entropysink.com

    there are two cardinal sins from which all others spring: Impatience and Laziness. - franz kafka

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    The Earth is not flat. Clyde's Avatar
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    Ah i see, i got it to be -x^2(x^2+1)^-3/2 + (x^2 + 1)^-1/2 which is unsimplified like you said.

    Normally when i have to use calculus it's to calculate numerical answers so i don't have to simplify stuff. I better practise my algebra evidently its getting rather rusty

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    Registered User whistlenm1's Avatar
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    f(x) = (x^2 + 1)^(1/2)

    f'(x) = x(x^2 + 1)^(-1/2)

    f''(x) = [(x^2 + 1)^(1/2) - x^2(X^2 + 1)^(-1/2)]/[((x^2 + 1)^(1/2))^2]

    = (x^2 + 1)^(-1/2) * [((x^2 + 1) - X^2)/(x^2 + 1)

    = (x^2 + 1)^(-1/2) * [1/(x^2 + 1)]

    or

    = 1/(x^2 +1)^(1/2) *1/(X^2 + 1)^(1)

    = 1/(x^2 + 1)^(3/2) or (x^2 + 1)^(-3/2)

    I believe
    Last edited by whistlenm1; 10-23-2003 at 10:10 AM.
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    Math is tricky and requires an attentive eye. I was wondering why I had come up with a different solution until I noticed I had missed a second x:
    Code:
    f''(x) = [(x^2 +1)^(-1/2)] -(1/2)x[(x^2 +1)^(-3/2)](2x)
    
    f''(x) != 1/[(x^2 +1)^(1/2)] -x/[(x^2 +1)(x^2 +1)^(1/2)]
    f''(x) != (x^2 -x +1)/[(x^2 +1)(x^2 +1)^(1/2)]
    
    f''(x) = 1/[(x^2 +1)(x^2 +1)^(1/2)]
    I am a programmer. My first duty is to God, then to nation, then to employer, then to family, then to friends, then to computer, and finally to myself. I code with dignity, honor, and integrity.

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    Unregistered Leeman_s's Avatar
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    What year in school did all of you take calculus? I'm taking it now...

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    Registered User axon's Avatar
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    Originally posted by Leeman_s
    What year in school did all of you take calculus? I'm taking it now...
    I took it first semester university.

    some entropy with that sink? entropysink.com

    there are two cardinal sins from which all others spring: Impatience and Laziness. - franz kafka

  10. #10
    ¡Amo fútbol!
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    Originally posted by Leeman_s
    What year in school did all of you take calculus? I'm taking it now...
    Junior

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    Pursuing knowledge confuted's Avatar
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    Originally posted by Leeman_s
    What year in school did all of you take calculus? I'm taking it now...
    I took it junior year, and I'm taking it again senior year (AP Calc BC now... before it was just the first 7 chapters of the calc book)
    Away.

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    Registered User whistlenm1's Avatar
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    a little over a year, and I have been tutoring eversince off and on again. The nightmares are gone but now I'm left with calc dreams!
    Man's mind once streched by a new idea, never regains its original dimensions
    - Oliver Wendell Holmes

    In other words, if you teach your cat to bark (output) and eat dog food (input) that doesn't make him a dog. It would have to chase cars, chew bones, and have puppies before I'd call it Rover ;-)
    - WaltP

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    Unregistered Leeman_s's Avatar
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    Wow, at my HS only four people are taking calc right now that are juniors. We must have on average stupid people here where I live

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    I'm taking Honors Advanced Precalc (as opposed to just normal or honors regular) as a Junior. I dunno if that counts .

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    Registered User axon's Avatar
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    Originally posted by Speedy5
    I'm taking Honors Advanced Precalc (as opposed to just normal or honors regular) as a Junior. I dunno if that counts .
    sorry, it doesn't... thats a course similar to what I took as highest in high school...now that I look back on it, it was nothing more than a waste of my time...

    some entropy with that sink? entropysink.com

    there are two cardinal sins from which all others spring: Impatience and Laziness. - franz kafka

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