Partial derivative of a non-defined function. (These are two different questions, not the same one) ![enter image description here]( Please forgive me, I'm not well-versed in using the symbols yet.

Hints:

For the first one,

$z_u = z_xx_u + z_yy_u = z_xe^u - z_ye^{-u}$
$z_v = -z_xe^{-v}-z_ye^v$

Now perform $z_u-z_v$

* * *

For the second one,

$$u=u\bigg(\frac 1x-\frac 1y,\frac1x - \frac1z\bigg)$$ Let $p = \frac 1x-\frac 1y,q=\frac1x - \frac1z$

$u_x = u_pp_x + u_qq_x$
$u_y = u_pp_y+u_qq_y$
$u_z = u_pp_z+u_qq_z$