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Implicit differentiation of one-parameter family

  1. Jan 13, 2008 #1
    1. The problem statement, all variables and given/known data

    Use implicit differentiation to show that the one parameter family f(x, y)=c satisfies the differential equation dy/dx = [tex]-f_{x}/f_{y} [/tex], where [tex]f_{x}=\frac{\partial f}{\partial x} [/tex] and [tex]f_{y}=\frac{\partial f}{\partial y} [/tex].

    2. Relevant equations

    3. The attempt at a solution

    Well, my teacher said I need to use the chain rule, but I'm confused about how to differentiate something that is in the general form f(x, y). And if f(x, y)=c, doesn't the derivative trivially equal 0?

    Thanks in advance for the help.
  2. jcsd
  3. Jan 13, 2008 #2


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    From the chain rule for the total derivative with respect to x:
    [tex]\frac{\partial f}{\partial x}\frac{dx}{dx} +\frac{\partial f}{\partial y}\frac{dy}{dx} = 0 [/tex]

    [tex]\frac{\partial f}{\partial x} +\frac{\partial f}{\partial y}\frac{dy}{dx} = 0 [/tex]

    Solving for dy/dx gives -fx/fy.
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