I have two examples of implicit differentiation:
Example 1:
$x^2+y^2+z^2-6=0$;
$xyz+2=0$
Supposing the both $x$ and $y$ are differentiable functions of $z$, then to compute $\frac{dx}{dz}$ and $\frac{dy}{dz}$ I get:
$$(1)\space\space\space\space 2x\frac{dx}{dz}+2y\frac{dy}{dz}+2z=0$$ $$(2)\space\space\space\space yz\frac{dx}{dz}+xz\frac{dy}{dz}+xy=0$$
How should I approach for solving the implicit differentiation, as I cannot manage to figure it out as I have yet been taught on how to do it, the answer should be for $\frac{dx}{dz}=\frac{x(y^2-z^2)}{z(x^2-y^2)}$?
Then there's also example two which confuses me even further:
$xu+yv+zw=1$;
$x+y+z+u+v+w=0$;
$xy+zuv+w=1$
Supposing that each of $x, y, z$ is a function of $u, v, w$, and differentiating in this case with respect to $w$, I can get linear system:
$$\begin{pmatrix}u & v & w \\ 1 & 1 & 1 \\ y & x & uv \end{pmatrix}\begin{pmatrix}x_w \\ y_w \\ z_w \end{pmatrix}=\begin{pmatrix}-z \\ -1 \\ -1 \end{pmatrix}$$
And solving this should yield:
$$\frac{\partial x}{\partial w} = \frac{uv^2+xz+w-zuv-xwv-v}{u^2v+vy+wx-yw-ux-uv^2}$$
I have thought of converting the left matrix into determinant form which gives me:
$$u^2v-ux-uv^2-vy+wx-wy$$
Which gives some aspects of the solution.
I do apologise as I have been staring at this for an hour and have not been getting close to the solution, your heartening help will be greatly appreciated!