Question:hard

If $z=f(x,y)$, $x=e^u+e^{-v}$, and $y=e^{-u}-e^v$, then:

Show Hint

To solve chain-rule equations quickly, write out the derivatives of intermediate variables first. Grouping terms systematically saves calculation time.

Updated On: Jul 24, 2026

$\frac{\partial z}{\partial u} - \frac{\partial z}{\partial v} = x \frac{\partial z}{\partial x} - y \frac{\partial z}{\partial y}$
$\frac{\partial z}{\partial u} + \frac{\partial z}{\partial v} = x \frac{\partial z}{\partial x} - y \frac{\partial z}{\partial y}$
$\left(\frac{\partial z}{\partial u} + \frac{\partial z}{\partial v}\right)^2 = x \frac{\partial z}{\partial x} - y \frac{\partial z}{\partial y}$
$\left(\frac{\partial z}{\partial u} - \frac{\partial z}{\partial v}\right)^2 = x \frac{\partial z}{\partial x} - y \frac{\partial z}{\partial y}$

Show Solution