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Math Help - Higher-order derivatives (simple problem)

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    Higher-order derivatives (simple problem)

    I am having trouble finding the second derivative of this function. A step by step solution would be greatly appreciated.

    f(x) = 3(2-x^2)^3
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    Quote Originally Posted by Archduke01 View Post
    I am having trouble finding the second derivative of this function. A step by step solution would be greatly appreciated.

    f(x) = 3(2-x^2)^3
    y = f(x) = 3(2 - x^2)^3.


    Let u = 2 - x^2 so that y = 3u^3.


    \frac{du}{dx} = -2x

    \frac{dy}{du} = 9u^2

     = 9(2 - x^2)^2.


    Therefore f'(x) = \frac{dy}{dx} = -18x(2 - x^2)^2.


    Now f''(x) = \frac{d}{dx}\left(\frac{dy}{dx}\right)

     = -18x\frac{d}{dx}[(2 - x^2)^2] + (2 - x^2)^2\frac{d}{dx}(-18x)

     = -18x(-2x)(2)(2 - x^2) - 18(2 - x^2)^2

     = 72x^2(2 - x^2) - 18(2 - x^2)^2

     = 18(2-x^2)[4x^2 - (2 - x^2)]

     = 18(2-x^2)(3x^2 - 2).
    Last edited by Prove It; November 13th 2009 at 07:35 PM.
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    Quote Originally Posted by Prove It View Post
    y = f(x) = 3(2 - x^2)^3.


    Let u = 2 - x^2 so that y = 3u^3.


    \frac{du}{dx} = -2x

    \frac{dy}{du} = 9u^2

     = 9(2 - x^2)^3.


    Therefore f'(x) = \frac{dy}{dx} = -18x(2 - x^2)^3.

    I'm grateful for your post, but I think you made a mistake. Shouldn't it be f'(x) = \frac{dy}{dx} = -18x(2 - x^2)^2?

    BTW, the answer is 18(2-x^2)(5x^2 - 2). I'm not sure where you went wrong with your method.

    Here's what I did;
    y' = 9(2-x^2)^2 (-2x)
    y'' = 18(2-x^2)(-2x)(-2)

    At this point I realized my method wasn't matching up to the given answer so I stopped. Could somebody tell me where I went wrong?
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    Quote Originally Posted by Archduke01 View Post
    I'm grateful for your post, but I think you made a mistake. Shouldn't it be f'(x) = \frac{dy}{dx} = -18x(2 - x^2)^2?

    BTW, the answer is 18(2-x^2)(5x^2 - 2). I'm not sure where you went wrong with your method.

    Here's what I did;
    y' = 9(2-x^2)^2 (-2x)
    y'' = 18(2-x^2)(-2x)(-2)

    At this point I realized my method wasn't matching up to the given answer so I stopped. Could somebody tell me where I went wrong?
    Yes you are right. And since my first derivative was wrong, so is my second derivative. Will edit now.
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    Quote Originally Posted by Prove It View Post
     = -18x\frac{d}{dx}[(2 - x^2)^2] + (2 - x^2)^2\frac{d}{dx}(-18x)

    I'm with you up until before the plus sign. How did you get everything after it?
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    Quote Originally Posted by Archduke01 View Post
    I'm with you up until before the plus sign. How did you get everything after it?
    It's a combination of the product chain rules.
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    Quote Originally Posted by Prove It View Post

     = -18x(-2x)(2)(2 - x^2) - 18(2 - x^2)^2

     = 72x(2 - x^2) - 18(2 - x^2)^2
    Shouldn't that be a 72x^2?

    EDIT: Nevermind, I understand the procedure now - thank you very much!
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    Quote Originally Posted by Archduke01 View Post
    Shouldn't that be a 72x^2?

    EDIT: Nevermind, I understand the procedure now - thank you very much!
    Yes it should be. Editing again.
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