Answer
$G_{x}=ze^{xz}sin(\frac{y}{z})$,
$G_{y}=\frac{e^{xz} cos(\frac{y}{z})}{z}$
and
$G_{z}=xe^{xz}sin(\frac{y}{z})-\frac{ye^{xz}cos(\frac{y}{z})}{z^{2}}$
Work Step by Step
Given: $G(x,y,z)=e^{xz}sin(\frac{y}{z})$
We need to find the first partial derivatives $G_{x}$,$G_{y}$ and $G_{z}$
Differentiate the function with respect to $x$ keeping $y$ and $z$ constant.
$G_{x}=ze^{xz}sin(\frac{y}{z})$
Differentiate the function with respect to $y$ keeping $x$ and $z$ constant.
$G_{y}=e^{xz}\times cos(\frac{y}{z}) \times \frac{1}{z}=\frac{e^{xz} cos(\frac{y}{z})}{z}$
Differentiate the function with respect to $z$ keeping $x$ and $y$ constant.
Apply product rule.
$G_{z}=xe^{xz}sin(\frac{y}{z})+e^{xz}\times cos(\frac{y}{z})\times \frac{-y}{z^{2}}$
$G_{z}=xe^{xz}sin(\frac{y}{z})-\frac{ye^{xz}cos(\frac{y}{z})}{z^{2}}$
Hence, $G_{x}=ze^{xz}sin(\frac{y}{z})$,
$G_{y}=\frac{e^{xz} cos(\frac{y}{z})}{z}$
and
$G_{z}=xe^{xz}sin(\frac{y}{z})-\frac{ye^{xz}cos(\frac{y}{z})}{z^{2}}$
