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

1 Answer
Mar 8, 2018

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