Implicit Function Theorem

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

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