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Implicit Differentiation problem

  1. Jul 9, 2007 #1
    1. The problem statement, all variables and given/known data

    Find dy/dx by implicit differentiation when it is known that y^2 + xsiny = 4

    2. Relevant equations

    3. The attempt at a solution

    2y dy/dt + xcosy dy/dt + siny = 0

    2y dy/dt + xcosy dy/dt = -siny

    dy/dt + dy/dt = -siny/2y/xcosy

    I'm sure I'm doing it wrong so I stopped right there. What am I doing it wrong and how do I solve it?

  2. jcsd
  3. Jul 9, 2007 #2
    you're looking for dy/dx not dy/dt o.o, but assuming by dt you mean dx then in the 2nd line of your attempt you should pull out the dy/dt and the answer will be easy from there.
  4. Jul 9, 2007 #3
    Is this the answer?

    dy/dx = -siny/2y+xcosy
  5. Jul 9, 2007 #4


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    It would be if you would parenthesize the denominator.
  6. Jul 9, 2007 #5
    dy/dx = -siny/(2y+xcosy) ? What's that's do
  7. Jul 9, 2007 #6


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    It distinguishes between -(siny/(2y))+(xcosy) and -siny/(2y+xcosy). Which are two quite different expressions. But look the same if you omit the parentheses.
    Last edited: Jul 9, 2007
  8. Jul 9, 2007 #7
    [tex]\frac{d}{dx}(y^2+x\sin(y)) = 0[/tex]

    [tex]=2y\left(\frac{dy}{dx}\right)+\left[\frac{d}{dx}x\cdot\sin(y)+\frac{d}{dx}\sin(y)\cdot x\right]=0[/tex]

    [tex]= 2y\left(\frac{dy}{dx}\right) + \left[\sin(y) + x\cdot\cos(y)\left(\frac{dy}{dx}\right)\right]= 2y\left(\frac{dy}{dx}\right) + \sin(y) + x\cos(y)\left(\frac{dy}{dx}\right) = 2y\left(\frac{dy}{dx}\right) + x\cos(y)\left(\frac{dy}{dx}\right) + \sin(y) = 0[/tex]

    [tex]= \left(\frac{dy}{dx}\right)(2y+x\cos(y)) + \sin(y) = 0[/tex]

    [tex]= \left(\frac{dy}{dx}\right)(2y+x\cos(y)) = -\sin(y)[/tex]

    [tex]\left(\frac{dy}{dx}\right) = \frac{-\sin(y)}{(2y+x\cos(y))}[/tex]

    Edit: I'm late by about 25 minutes. :biggrin:
  9. Jul 9, 2007 #8
    cool thanks
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