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Implicit differentiation problem.

  1. Apr 28, 2012 #1
    1. The problem statement, all variables and given/known data

    Find dy/dx in terms of x and y if..

    x2-√(xy)+y2=6





    2. Relevant equations



    3. The attempt at a solution

    so I started by..


    x2-√(xy)+y2=6

    deriving the LHS

    2x+2y(dy/dx)-1/2(xy)-1/2(1(y)+x(dy/dx))

    Simplifying the last term

    2x+2y(dy/dx)-(y+x(dy/dx))/(2√(xy))=6

    taking the 2x over to seperate dy/dx's

    2y(dy/dx)-(y+x(dy/dx))/(2√(xy))= 6-2x

    then I thought I would multiply through to get a common denominator..

    [2y(dy/dx)(2√(xy))-(y+x(dy/dx))] / [2√(xy) =6 -2x

    so multiply through by denominator to simplify and collect like terms

    [2y(dy/dx)(2√(xy))-(y+x(dy/dx))] = (6-2x)(2√(xy))

    so taking that -y over

    [2y(dy/dx)(2√(xy))-(x(dy/dx))]= (6-2x)(2√(xy))+y

    taking out dy/dx as a common factor

    dy/dx[4y√(xy) - x] = (6-2x)(2√(xy))+y

    so dy/dx = [(6-2x)(2√(xy))+y] / [4y√(xy) - x]

    Is this right, because I checked it on wolfram and it had a different answer so I guess not, can someone please shed some light on where I went wrong?

    Thanks.
     
  2. jcsd
  3. Apr 28, 2012 #2

    Dick

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    Homework Helper

    After the step labelled "Simplifying the last term" why to you have 6 on the right side?
     
  4. Apr 28, 2012 #3
    Left hand side is good, but I'm not so sure you want to simplify the last term. What is ##\frac{d}{dx} (6)##?

    Every term on the left hand side isn't multiplied by ##\frac{dy}{dx}##, you need to move the other term over as well before you can factor ##\frac{dy}{dx}## out.

    Try fixing those errors and continue.
     
  5. Apr 28, 2012 #4


    ok yeah the 6 will cancel.

    and when I take the 2x over I also take the -y over

    "" " " "" = y-2x

    ??
     
  6. Apr 28, 2012 #5
    I'm not sure what you mean by the 6 canceling. The derivative of a constant is 0.

    This just becomes an algebra exercise from here. The y isn't a term by itself, it is being multiplied by a number.

    Don't simplify the last term, multiply the last term out and then gather all the terms with ##\frac{dy}{dx}## onto one side and all the other terms onto the other side.
     
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