An example- to aid in thinking about Implicit Differentiation

F[x, y, z]=0 determines a surface in space. In this case it contains the solutions to the equation y/x=Tan(z)

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We can parameterize this surface as follows:

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Then, we see the image of P is sent to zero by F

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We can compute the partial derivatives of F with respect to x and y and z

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Using implicit differentiation we determine the partials dz/dx and dz/dy

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Note, those equations were not defined on the line x=y=0 which lies on our surface F[x,y,z]=0

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We can use the partials to compute the equation of the tangent plane at the point (x,y,z)

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Here is our surface again

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Here is the tangent plane at the point (1,0,0)

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Let's see what happens to the tangent plane at (x,0,0) as x->0. (Note the partials at (0,0,0) were not defined))

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As (x,0,0)->(0,0,0) the planes become closer and closer to vertical! Think about why these partial derivatives of a vertical plane are not well defined

Created by Mathematica  (October 5, 2004)