The task is to find $\frac{\partial w}{\partial x}$ of the function $w = zxe^y + xe^z + ye^z$ with the constraint $x^2y + xy^2 = 1$. Assume $x$ is independent.
So, we first find the differential of the constraint to be $(2x + y^2)dx + (x^2 + 2xy)dy = 0$, from which $dy = \frac{2xy + y^2}{x^2 + 2xy}dx$.
Then, substituting this into the total differential of $w$, we get $dw = (ze^y + e^z)dx + (zxe^y + e^z)dy + (xe^y + xe^z + ye^z)dz$. Key question now:
The latter expression then looks as the following $dw = (0 + e^0)dx + (0 + e^0)(\frac{2xy + y^2}{x^2 + 2xy}) + (xe^y + xe^z + ye^z)$ which after algebraic transformations gives the final $\frac{\partial w}{\partial x} = \frac{x^2 + 4xy + y^2}{x^2 + 2xy}$
Why do we make $z = 0$ in the expression for $dw$ above? Is this because it is not in the constraint? So, we don't look at the $xz, yz-$planes, just the $xy-$plane?