Answer
$$\frac{{\partial w}}{{\partial x}} = \frac{{2x}}{{{y^2} + {z^2}}}$$
$$\frac{{\partial w}}{{\partial y}} = \frac{{ - 2y\left( {{z^2} + {x^2}} \right)}}{{{{\left( {{y^2} + {z^2}} \right)}^2}}}$$
$$\frac{{\partial w}}{{\partial z}} = \frac{{2z\left( {{y^2} - {x^2}} \right)}}{{{{\left( {{y^2} + {z^2}} \right)}^2}}}$$
Work Step by Step
$$\eqalign{
& w = \frac{{{x^2} - {y^2}}}{{{y^2} + {z^2}}} \cr
& {\text{Calculate }}\frac{{\partial w}}{{\partial x}} \cr
& \frac{{\partial w}}{{\partial x}} = \frac{\partial }{{\partial x}}\left[ {\frac{{{x^2} - {y^2}}}{{{y^2} + {z^2}}}} \right] \cr
& \frac{{\partial w}}{{\partial x}} = \frac{\partial }{{\partial x}}\left[ {\frac{{{x^2}}}{{{y^2} + {z^2}}}} \right] - \frac{\partial }{{\partial x}}\left[ {\frac{{{y^2}}}{{{y^2} + {z^2}}}} \right] \cr
& \frac{{\partial w}}{{\partial x}} = \frac{1}{{{y^2} + {z^2}}}\frac{\partial }{{\partial x}}\left[ {{x^2}} \right] \cr
& \frac{{\partial w}}{{\partial x}} = \frac{{2x}}{{{y^2} + {z^2}}} \cr
& \cr
& {\text{Calculate }}\frac{{\partial w}}{{\partial y}} \cr
& \frac{{\partial w}}{{\partial y}} = \frac{\partial }{{\partial y}}\left[ {\frac{{{x^2} - {y^2}}}{{{y^2} + {z^2}}}} \right] \cr
& {\text{use quotient rule}} \cr
& \frac{{\partial w}}{{\partial y}} = \frac{{\left( {{y^2} + {z^2}} \right)\left( { - 2y} \right) - \left( {{x^2} - {y^2}} \right)\left( {2y} \right)}}{{{{\left( {{y^2} + {z^2}} \right)}^2}}} \cr
& \frac{{\partial w}}{{\partial y}} = \frac{{ - 2{y^3} - 2y{z^2} - 2{x^2}y + 2{y^3}}}{{{{\left( {{y^2} + {z^2}} \right)}^2}}} \cr
& \frac{{\partial w}}{{\partial y}} = \frac{{ - 2y{z^2} - 2{x^2}y}}{{{{\left( {{y^2} + {z^2}} \right)}^2}}} \cr
& \frac{{\partial w}}{{\partial y}} = \frac{{ - 2y\left( {{z^2} + {x^2}} \right)}}{{{{\left( {{y^2} + {z^2}} \right)}^2}}} \cr
& \cr
& {\text{Calculate }}\frac{{\partial w}}{{\partial z}} \cr
& \frac{{\partial w}}{{\partial z}} = \frac{\partial }{{\partial z}}\left[ {\frac{{{x^2} - {y^2}}}{{{y^2} + {z^2}}}} \right] \cr
& \frac{{\partial w}}{{\partial z}} = \left( {{x^2} - {y^2}} \right)\frac{\partial }{{\partial z}}\left[ {{{\left( {{y^2} + {z^2}} \right)}^{ - 1}}} \right] \cr
& \frac{{\partial w}}{{\partial z}} = - \left( {{x^2} - {y^2}} \right){\left( {{y^2} + {z^2}} \right)^{ - 2}}\frac{\partial }{{\partial z}}\left[ {{y^2} + {z^2}} \right] \cr
& \frac{{\partial w}}{{\partial z}} = - 2z\left( {{x^2} - {y^2}} \right){\left( {{y^2} + {z^2}} \right)^{ - 2}} \cr
& \frac{{\partial w}}{{\partial z}} = \frac{{2z\left( {{y^2} - {x^2}} \right)}}{{{{\left( {{y^2} + {z^2}} \right)}^2}}} \cr} $$
