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

At this point we investigate the sign of $\sqrt {x^2 - v^2}$ which is needed.

We see that:

Hence when we are at the point of plugging in numbers we will need to take the positive square root.

Then we have:

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