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

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Steve M Share
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 $F \left(x , y\right) = 0$ which defines $y$ as a function of $x$ and $y$. Therefore we can write $F \left(x , y\right) = 0$ as $F \left(x , y \left(x\right)\right) = 0$. 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:

$2 z + 2 y \frac{\mathrm{dy}}{\mathrm{dx}} = 0 \implies \frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{x}{y}$

And if we write this in the form:

$F \left(x , y\right) = 0 \implies F \left(x , y\right) = - {x}^{2} + {y}^{2} - 1$

Then the partial derivatives are:

${F}_{=} x = 2 x$ and ${F}_{=} y = 2 y$

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

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