Partial Derivative/Examples/2 u + 3 v = sin x, u + 2 v = x cos y/Implicit Method

Proof
By definition of partial derivative:


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

hence the motivation for the abbreviated notation on the.

and so combining $(1)$ and $(2)$ into matrix form:


 * $\begin {pmatrix} 2 & 3 \\ 1 & 2 \end {pmatrix} \begin {pmatrix} \dfrac {\partial u} {\partial x} \\ \dfrac {\partial v} {\partial x} \end {pmatrix} = \begin {pmatrix} \cos x \\ \cos y \end {pmatrix}$

Hence by Cramer's Rule:

Hence: