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

Example of Partial Derivative
Consider the simultaneous equations:


 * $\begin {cases} 2 u + 3 v & = \sin x \\ u + 2 v & = x \cos y \end {cases}$

Then:


 * $\map {u_1} {\dfrac \pi 2, \pi} = 3$

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.