Partial Derivative/Examples/2 u + 3 v = sin x, u + 2 v = x cos y/Explicit 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.

We have:
 * $2 u + 3 v = \sin x$

Thus:

Then we have:
 * $u + 2 v = x \cos y$

Thus:

Substituting $\dfrac {\partial v} {\partial x}$ from $(2)$ into $(1)$ gives: