Answer
$$
e^{z}=xyz
$$
To find $\frac{\partial z}{\partial x}$, we differentiate implicitly with respect to x, being careful to treat $y$ as a constant:
$$
\begin{aligned}
\frac{\partial z}{\partial x} &= \frac{ yz}{(e^{z}- y x )}.
\end{aligned}
$$
To find $\frac{\partial z}{\partial y}$, we differentiate implicitly with respect to $y$, being careful to treat $x$ as a constant:
$$
\begin{aligned}
\frac{\partial z}{\partial y} &= \frac{ xz}{(e^{z}- xy )}.
\end{aligned}
$$
Work Step by Step
$$
e^{z}=xyz
$$
To find $\frac{\partial z}{\partial x}$, we differentiate implicitly with respect to x, being careful to treat $y$ as a constant:
$$
\begin{aligned}
\frac{\partial }{\partial x} (e^{z}) &=\frac{\partial }{\partial x} (xyz)\\
e^{z}\frac{\partial z}{\partial x} &= y (x \frac{\partial z }{\partial x}+z.1 )
\\
e^{z}\frac{\partial z}{\partial x} &= y x \frac{\partial z }{\partial x}+ yz
\\
e^{z}\frac{\partial z}{\partial x} - y x \frac{\partial z }{\partial x} &= yz
\\
\frac{\partial z}{\partial x} (e^{z}- y x )&= yz\\
\\
\frac{\partial z}{\partial x} &= \frac{ yz}{(e^{z}- y x )}
\end{aligned}
$$
To find $\frac{\partial z}{\partial y}$, we differentiate implicitly with respect to $y$, being careful to treat $x$ as a constant:
$$
\begin{aligned}
\frac{\partial }{\partial y} (e^{z}) &=\frac{\partial }{\partial y} (xyz)\\
e^{z}\frac{\partial z}{\partial y} &= x (y \frac{\partial z }{\partial y}+z.1 )
\\
e^{z}\frac{\partial z}{\partial y} &= x y \frac{\partial z }{\partial y}+ xz
\\
e^{z}\frac{\partial z}{\partial y} - x y \frac{\partial z }{\partial y} &= xz
\\
\frac{\partial z}{\partial y} (e^{z}- xy )&= xz\\
\\
\frac{\partial z}{\partial y} &= \frac{ xz}{(e^{z}- xy )}
\end{aligned}
$$
