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)

F[{x_, y_, z_}] := x Sin[z] - y Cos[z]

F[{x, y, z}]

We can parameterize this surface as follows:

P[t_, z_] := {t*Cos[z], t*Sin[z], z}

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

F[P[t, z]]

ParametricPlot3D[P[t, z], {t, -2, 2}, {z, -Pi, Pi}, 
      AspectRatio -> Automatic]

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

Fx = D[F[{x, y, z}], x]
Fy = D[F[{x, y, z}], y]
Fz = D[F[{x, y, z}], z]

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

dzdx = -Fx/Fz
dzdy = -Fy/Fz

dzdx2[x_, y_, 
        z_] := -Sin[z]/(x Cos[z] + 
              y Sin[z])
dzdy2[x_, y_, z_] := Cos[z]/(x Cos[z] + y Sin[z])

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

F[{0, 0, z}]

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

T[x_, y_, z_][a_, b_] := 
      z + (a - x) dzdx2[x, y, z] + (b - y) dzdy2[x, y, z]

Here is our surface again

g1 = 
      ParametricPlot3D[P[t, z], {t, -2, 2}, {z, -Pi, Pi}, 
        AspectRatio -> Automatic]

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

g2 = 
      Plot3D[T[1, 0, 0][a, b], {a, -2, 2}, {b, -2, 2}]

Show[{g1, g2}]

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))

tanplane[n_] := 
      Plot3D[T[n, 0, 0][a, b], {a, -2, 2}, {b, -2, 2}]

tab = Table[tanplane[2/i], {i, 1, 10}]

For[i = 1, i < 10, i++, Show[g1, tab[[i]]]]

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