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Implicit Function Theorem


Suppose that at the point (a, b), z = F(x, y) is smooth, F(a, b) = 0, and ∂z/∂y ≠ 0.

Then the curve F(x, y) = 0 at (a, b) has an implicit function and the slope


There are three things to prove:

(1)    There exists an implicit function y = g(x) at (a, b).

(2)    The slope dy/dx = g'(a) exists.

(3)    dy/dx has the required value.

Instead of proving the whole theorem, we give an intuitive argument for (1) and (2) and then prove (3).

The surface z = F(x, y) has a tangent plane at (a, b, 0). If we intersect the surface and tangent plane with the plane z = 0 we get the curve 0 = F(x, y) and a line L. Through an infinitesimal microscope aimed at the point (a, b), the curve looks like the graph of a function y = g(x) which has the tangent line L and thus has a slope at (a, b) (Figure 11.6.4).


Figure 11.6.4


Given that the slope 11_partial_differentiation-370.gif exists, we compute its value.

By the Chain Rule,


But F(x, g(x)) is identically zero, so dz/dx = 0 and




The best way to remember the minus sign in the above equation is to derive the equation yourself. Start with the Chain Rule for dz/dx = 0 and solve for dy/dx. One way to understand the minus sign is as follows: if ∂z/∂x and ∂z/∂y are positive, an increase in x must be offset by a decrease in y to keep z constant, so dy/dx should be negative.

Warning: The two ∂z's have different meanings and cannot be cancelled.

Example 3: Slope

The Implicit Function Theorem gives us a convenient equation for the tangent line to the curve F(x, y) = 0 at (a, b).


and finally

Tangent Line:


Example 3 (Continued): Tangent Line
Example 4: Tangent Line and Slope
Example 5: Same of Hyperbolic Paraboloid

Last Update: 2006-11-05