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Implicit differentiation of one-parameter family

  • Thread starter bleucat
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1. Homework Statement

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. Homework 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.
 

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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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