Solving for implicit functions of $x$ and $y$

Question

Given $2x = u^{2}-v^{2}$ and $y=uv$, where $u$ and $v$ are implicit functions of $x$ and $y$.

I am asked to find $\frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial v}{\partial x}, \frac{\partial v}{\partial y}$.

By the implicit function theorem, $u^{2}-v^{2}-2x = 0 \equiv F$ and $uv - y = 0 \equiv G$

So by the implicit function rule, we have $$\frac{\partial u}{\partial x} = - \frac{\partial F/\partial x}{\partial F/\partial u} = \frac{1}{u}$$.

Similarly, we have $$\frac{\partial u}{\partial y} = \frac{1}{v},$$ $$\frac{\partial v}{\partial x}=\frac{-1}{v},$$ $$\frac{\partial v}{\partial y}=\frac{1}{u}.$$

Are my solutions correct?

Answer 1

I don't think that your method is correct.

Compare to the direct partial derivation method (below) and see where is the mistake.

Answer 2

The problem is your formulas for the derivatives, since your functions F and G depend on 4 variables, 2 independent and 2 dependent ones, the derivatives should be like this:

$$\left(\begin{array}{cc}\partial u/\partial x & \partial u/\partial y \\ \partial v/\partial x & \partial v/\partial x\end{array}\right) = - \left(\begin{array}{cc} \partial F/\partial u & \partial F/\partial v \\ \partial G/\partial u & \partial G/\partial v\end{array}\right)^{-1} \left(\begin{array}{cc} \partial F/\partial x & \partial F/\partial y \\ \partial G/\partial x & \partial G/\partial y \end{array}\right) $$

which if you work it out should indeed give you what JJacquelin calculated.