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Implicit Differentiation Problem - Check my work?

  1. Oct 2, 2005 #1
    Implicit Differentiation Problem -- Check my work?

    I've worked it -- can someone just check my work?

    Problem:

    xcosy+ycos=1

    My work:

    [x (d/x)cosy + cosy (d/dx)x] + [y (d/dx)cosx + cosx (d/dx)y] = (d/dx) 1

    -xsiny (dy/dx) + cos y - ysinx + cos x (dy/dx) = 0

    -xsiny (dy/dx) + cos y = ysinx - cosy

    dy/dx = (ysinx - cosy)/(-xsiny + cos x)


    Meanwhile, could someone help me with this one...

    squareroot (xy) = 1+(x^2)y
     
    Last edited: Oct 2, 2005
  2. jcsd
  3. Oct 4, 2005 #2

    kreil

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    Gold Member

    the first one looks good, as for the second:



    (2)[tex]xy=1+x^2y[/tex]

    You are differentiating WRT y, which means that the derivative of x is 1, but the derivative of y is dy/dx. Make sure you use the product rule:

    [tex]x\frac{dy}{dx}+y=x^2 \frac{dy}{dx}+2xy[/tex]

    [tex]x\frac{dy}{dx}-x^2 \frac{dy}{dx}=2xy-y[/tex]

    [tex]\frac{dy}{dx}(x-x^2)=2xy-y[/tex]

    [tex]\frac{dy}{dx}=\frac{2xy-y}{x-x^2}[/tex]

    ~Josh
     
  4. Oct 4, 2005 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    You miswrote the first line above but obviously that was a typo since you got it right in the end. If I were your teacher I would prefer to see one more line:
    -xsiny (dy/dx) + cos y = ysinx - cos x(dy/dx)

    cos x (dy/dx)- x sin y (dy/dx)= (cos x- x sin y)(dy/dx)= y sin x- cos y

    dy/dx = (ysinx - cosy)/(-xsiny + cos x)
     
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