# The Implicit Function Theorem Examples 1

Be sure to review The Implicit Function Theorem page before looking at the examples given below.

## Example 1

**Find $\left ( \frac{\partial x}{\partial y} \right )_u$ if $\left\{\begin{matrix} xyuv = 1\\ x + y + u + v = 0 \end{matrix}\right.$.**

Let $F(x, y, u, v) = xyuv – 1 = 0$ and $G(x, y, u, v) = x + y + u + v = 0$. The notation $\left ( \frac{\partial x}{\partial y} \right )_u$ implies that the variables $x$ and $v$ are dependent, while the variables $y$ and $u$ are independent.

(1)

Let’s compute these Jacobians. We have that the Jacobian on the numerator is:

(2)

The Jacobian in the denominator is:

(3)

Therefore we have that:

(4)

## Example 2

**Find $\left ( \frac{\partial x}{\partial y} \right )_z$ if $\left\{\begin{matrix} x^2 + y^2 + z^2 + w^2 = 1\\ x + 2y + 3z + 4w = 2 \end{matrix}\right.$.**

Let $F(x, y, z, w) = x ^2 + y^2 + z^2 + w^2 – 1 = 0$ and let $G(x, y, z, w) = x + 2y + 3z + 4w – 2 = 0$. The notation $\left ( \frac{\partial x}{\partial y} \right )_z$ implies that $x$ and $w$ are dependent variables and $y$ and $z$ are independent variables. We have that then:

(5)

Let’s compute these Jacobians. The Jacobian in the numerator is given by:

(6)

The Jacobian in the denominator is given by:

(7)

Therefore we have that:

(8)

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