As with many other problems like this the rule of choice is the chain rule:
$\displaystyle z = ln(zy + e^x)$
$\displaystyle z_y = \frac{1}{zy + e^x} \cdot \frac{\partial }{\partial y} \left ( zy + e^2 \right ) $
(You fill in the rest.)
After you are done, just replace the z with ln(zy + e^x).
If you need more help, just let us know.
-Dan