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Math Help - Derivative?

  1. #1
    Super Member fardeen_gen's Avatar
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    Derivative?

    If  y = \frac{\sin x}{1 + \frac{\cos x}{1 + \frac{\sin x}{1 + \frac{\cos x}{1 + \sin x\mbox{...}\infty}}}}, then prove that \frac{dy}{dx} = \frac{(1 + y)\cos x + y\sin x}{1 + 2y + \cos x - \sin x}
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  2. #2
    MHF Contributor alexmahone's Avatar
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    y=\frac{sin x}{1+\frac{cos x}{1+y}}
    y=\frac{sin x(1+y)}{1+y+cos x}
    y+y^2+ycos x=sin x+ysin x
    \frac{dy}{dx}+2y\frac{dy}{dx}+y*-sin x+cos x*\frac{dy}{dx}=cos x+ycos x+sin x\frac{dy}{dx}
    \frac{dy}{dx}(1+2y+cos x)-ysin x=cos x(1+y)+sin x\frac{dy}{dx}
    \frac{dy}{dx}(1+2y+cos x-sin x)=(1+y)cos x+y sin x
    \frac{dy}{dx}(1+2y+cos x-sin x)=(1+y)cos x+y sin x
    \frac{dy}{dx}=\frac{(1+y)cos x+y sin x}{1+2y+cos x-sin x}
    Last edited by alexmahone; June 10th 2009 at 08:35 PM.
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  3. #3
    Super Member Random Variable's Avatar
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    I'd like to see the calculus textbook from which you get these unusual (but interesting) problems.
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  4. #4
    Moo
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    Quote Originally Posted by Random Variable View Post
    I'd like to see the calculus textbook from which you get these unusual (but interesting) problems.
    See his profile, Isomorphism and I already asked this question lol
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  5. #5
    Super Member flyingsquirrel's Avatar
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    Quote Originally Posted by fardeen_gen View Post
     y = \frac{\sin x}{1 + \frac{\cos x}{1 + \frac{\sin x}{1 + \frac{\cos x}{1 + \sin x\mbox{...}\infty}}}}
    To write continued fractions one can use \cfrac instead of \frac, it makes the fraction more readable.

     y = \cfrac{\sin x}{1 + \cfrac{\cos x}{1 + \cfrac{\sin x}{1 + \cfrac{\cos x}{1 + \cfrac{\sin x}{1 + \cfrac{\cos x}{1 + \cfrac{\sin x}{1 + \cfrac{\cos x}{1 + \ldots}}}}}}}}
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  6. #6
    Super Member fardeen_gen's Avatar
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    Quote Originally Posted by flyingsquirrel View Post
    To write continued fractions one can use \cfrac instead of \frac, it makes the fraction more readable.


     y = \cfrac{\sin x}{1 + \cfrac{\cos x}{1 + \cfrac{\sin x}{1 + \cfrac{\cos x}{1 + \cfrac{\sin x}{1 + \cfrac{\cos x}{1 + \cfrac{\sin x}{1 + \cfrac{\cos x}{1 + \ldots}}}}}}}}
    I wanted to create a new thread asking exactly this - how do write continued fractions. Thanks a lot
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