Partial Derivative/Examples/u^2 + v^2 = x^2, 2 u v = 2 x y + y^2/Implicit Method

Proof
By definition of partial derivative:


 * $\map {u_1} {1, -2} = \valueat {\dfrac {\partial u} {\partial x} } {x \mathop = 1, y \mathop = -2}$

hence the motivation for the abbreviated notation on the.

Lemma
We have:

Then we have:

Combining $(1)$ and $(2)$ into matrix form:


 * $\begin {pmatrix} u & v \\ v & u \end {pmatrix} \begin {pmatrix} \dfrac {\partial u} {\partial x} \\ \dfrac {\partial v} {\partial x} \end {pmatrix} = \begin {pmatrix} x \\ y \end {pmatrix}$

Hence by Cramer's Rule:

Hence: