Answer
$$\frac{{\partial w}}{{\partial x}} = yz{e^z}\cos xz$$
$$\frac{{\partial w}}{{\partial y}} = {e^z}\sin xz$$
$$\frac{{\partial w}}{{\partial y}} = y{e^z}\left( {x\cos xz + \sin xz} \right)$$
Work Step by Step
$$\eqalign{
& w = y{e^z}\sin xz \cr
& {\text{Calculate }}\frac{{\partial w}}{{\partial x}} \cr
& \frac{{\partial w}}{{\partial x}} = \frac{\partial }{{\partial x}}\left[ {y{e^z}\sin xz} \right] \cr
& \frac{{\partial w}}{{\partial x}} = y{e^z}\frac{\partial }{{\partial x}}\left[ {\sin xz} \right] \cr
& \frac{{\partial w}}{{\partial x}} = y{e^z}\left( {z\cos xz} \right) \cr
& \frac{{\partial w}}{{\partial x}} = yz{e^z}\cos xz \cr
& \cr
& {\text{Calculate }}\frac{{\partial w}}{{\partial x}} \cr
& \frac{{\partial w}}{{\partial y}} = \frac{\partial }{{\partial y}}\left[ {y{e^z}\sin xz} \right] \cr
& \frac{{\partial w}}{{\partial y}} = {e^z}\sin xz\frac{\partial }{{\partial y}}\left[ y \right] \cr
& \frac{{\partial w}}{{\partial y}} = {e^z}\sin xz \cr
& \cr
& {\text{Calculate }}\frac{{\partial w}}{{\partial x}} \cr
& \frac{{\partial w}}{{\partial y}} = \frac{\partial }{{\partial y}}\left[ {y{e^z}\sin xz} \right] \cr
& {\text{use product rule}} \cr
& \frac{{\partial w}}{{\partial y}} = y{e^z}\frac{\partial }{{\partial y}}\left[ {\sin xz} \right] + \sin xz\frac{\partial }{{\partial y}}\left[ {y{e^z}} \right] \cr
& \frac{{\partial w}}{{\partial y}} = y{e^z}\left( {x\cos xz} \right) + \sin xz\left( {y{e^z}} \right) \cr
& \frac{{\partial w}}{{\partial y}} = y{e^z}\left( {x\cos xz + \sin xz} \right) \cr} $$
