Question

What is the Implicit Function Theorem and how do you prove it?

Answer 1

The Implicit Function Theorem is a method of using partial derivatives to perform implicit differentiation.

Suppose we cannot find `y` explicitly as a function of `x`, only implicitly through the equation `F(x, y) = 0` which defines `y` as a function of `x` and `y`. Therefore we can write `F(x, y) = 0` as `F(x, y(x)) = 0`. Differentiating both sides of this, using the partial chain rule gives us

`(partial F)/(partial x) dx/dx + (partial F)/(partial y) dy/dx = 0 => dy/dx = −((partial F)/(partial x)) / ((partial F)/(partial y))`

Example:
If we take a unit circle, with equation:

`x^2 + y^2 = 1`

Then differentiating implicitly we have:

`2z + 2ydy/dx = 0=> dy/dx= -x/y`

And if we write this in the form:

`F(x,y) = 0 => F(x,y) = -x^2 + y^2 - 1`

Then the partial derivatives are:

`F_=x = 2x` and `F_=y = 2y`

And the Implicit Function Theorem gives

`dy/dx = −F_x/F_y`

`\ \ \ \ \ \ = −((partial F)/(partial x)) / ((partial F)/(partial y))`

`\ \ \ \ \ \ = −(2x) / (2y)`

`\ \ \ \ \ \ = −x/y`, as before