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

Example of Partial Derivative
Consider the simultaneous equations:

Then:
 * $\map {u_1} {1, -2} = 1$

at $u = 1$, $v = 0$.

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.