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Implicit Differentiation

  1. Jul 9, 2009 #1
    1. The problem statement, all variables and given/known data

    Show that the set defined by the equations
    [tex]x + y + z + w = sin(xyzw)[/tex]

    [tex]
    x - y + z + w^2 = cos(xyzw) - 1[/tex]
    can be described explicitly by equation of the form (z, w) = f(x, y) near the point (0,0,0,0); find the first partial derivatives of f(x,y) at the point (0,0)

    2. Relevant equations

    The above bolded part is the part I'm unsure about...

    3. The attempt at a solution

    I did:
    [tex]
    G = x + y + z + w - sin(xyzw)[/tex]

    [tex]
    H = x - y + z + w^2 - cos(xyzw) + 1[/tex]

    [tex]
    \frac{\partial (G, H)}{\partial (x, y)} + \frac{\partial (G, H)}{\partial (z, w)} \frac{\partial f}{\partial x } = 0
    [/tex]

    Then I solved for [tex]\frac{\partial f}{\partial x}[/tex]?
     
  2. jcsd
  3. Jul 9, 2009 #2
    I think the note of "near the point (0,0,0,0)" is a clue that you can use the small angle approximation of

    [tex]
    \sin\theta = \theta,\quad

    \cos\theta = 1,\quad

    \texttt{when }\theta \texttt{ is small.}
    [/tex]

    I'm not fully understanding what the question is asking, but I'm interpreting it as saying you need to find

    [tex]
    z = f_1(x,y)
    [/tex]

    [tex]
    w = f_2(x,y)
    [/tex]

    From there, I would calculate
    [tex]
    \left.\frac{\partial z}{\partial x}\right|_{(0,0)} = \frac{\partial f_1(0,0)}{\partial x}
    [/tex]
    [tex]
    \left.\frac{\partial z}{\partial y}\right|_{(0,0)} = \frac{\partial f_1(0,0)}{\partial y}
    [/tex]
    [tex]
    \left.\frac{\partial w}{\partial x}\right|_{(0,0)} = \frac{\partial f_2(0,0)}{\partial x}
    [/tex]
    [tex]
    \left.\frac{\partial w}{\partial y}\right|_{(0,0)} = \frac{\partial f_2(0,0)}{\partial y}
    [/tex]

    As I said, though, I may not be properly understanding the question.
     
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