Answer
$$\eqalign{ & {w_s} = - \frac{{2t\left( {t + 1} \right)}}{{{{\left( {st + s - t} \right)}^2}}} \cr & {w_t} = \frac{{2s}}{{{{\left( {st + s - t} \right)}^2}}} \cr} $$
Work Step by Step
$$\eqalign{ & {\text{Let the functions }}w = \frac{{x - z}}{{y + z}},\,\,\,\,\,x = s + t,\,\,\,y = st\,\,\,{\text{ and }}z = s - t \cr & \cr & {\text{Calculate }}{z_s},{\text{ apply }}\frac{{\partial w}}{{\partial s}} = \frac{{\partial z}}{{\partial x}}\frac{{\partial x}}{{\partial s}} + \frac{{\partial z}}{{\partial y}}\frac{{\partial y}}{{\partial s}} + \frac{{\partial z}}{{\partial z}}\frac{{\partial z}}{{\partial s}} \cr & \frac{{\partial w}}{{\partial x}} = \frac{\partial }{{\partial x}}\left[ {\frac{{x - z}}{{y + z}}} \right] = \frac{1}{{y + z}} \cr & \frac{{\partial w}}{{\partial y}} = \frac{\partial }{{\partial y}}\left[ {\frac{{x - z}}{{y + z}}} \right] = \frac{{z - x}}{{{{\left( {y + z} \right)}^2}}} \cr & \frac{{\partial w}}{{\partial t}} = \frac{\partial }{{\partial y}}\left[ {\frac{{x - z}}{{y + z}}} \right] = - \frac{{x + y}}{{{{\left( {y + z} \right)}^2}}} \cr & \frac{{\partial x}}{{\partial s}} = \frac{\partial }{{\partial s}}\left[ {s + t} \right] = 1 \cr & \frac{{\partial y}}{{\partial s}} = \frac{\partial }{{\partial s}}\left[ {st} \right] = t \cr & \frac{{\partial z}}{{\partial s}} = \frac{\partial }{{\partial s}}\left[ {s - t} \right] = 1 \cr & \cr & {\text{Then}}{\text{,}} \cr & \frac{{\partial w}}{{\partial s}} = \left( {\frac{1}{{y + z}}} \right)\left( 1 \right) + \left( {\frac{{z - x}}{{{{\left( {y + z} \right)}^2}}}} \right)\left( t \right) + \left( { - \frac{{x + y}}{{{{\left( {y + z} \right)}^2}}}} \right)\left( 1 \right) \cr & \frac{{\partial w}}{{\partial s}} = \frac{1}{{y + z}} + \frac{{z - x}}{{{{\left( {y + z} \right)}^2}}} - \frac{{x + y}}{{{{\left( {y + z} \right)}^2}}} \cr & \frac{{\partial w}}{{\partial s}} = \frac{{y + z + \left( {z - x} \right)t - x - y}}{{{{\left( {y + z} \right)}^2}}} \cr & \frac{{\partial w}}{{\partial s}} = \frac{{y + z + tz - tx - x - y}}{{{{\left( {y + z} \right)}^2}}} \cr & {\text{Write in terms of }}s{\text{ and }}t \cr & \frac{{\partial w}}{{\partial s}} = \frac{{st + s - t + t\left( {s - t} \right) - t\left( {s + t} \right) - \left( {s + t} \right) - st}}{{{{\left( {st + s - t} \right)}^2}}} \cr & \frac{{\partial w}}{{\partial s}} = \frac{{st + s - t + ts - {t^2} - ts - {t^2} - s - t - st}}{{{{\left( {st + s - t} \right)}^2}}} \cr & \frac{{\partial w}}{{\partial s}} = \frac{{ - t - {t^2} - {t^2} - t}}{{{{\left( {st + s - t} \right)}^2}}} \cr & \frac{{\partial w}}{{\partial s}} = - \frac{{2t\left( {t + 1} \right)}}{{{{\left( {st + s - t} \right)}^2}}} \cr & \cr & {\text{Calculate }}{z_t},{\text{ apply }}\frac{{\partial z}}{{\partial t}} = \frac{{\partial z}}{{\partial x}}\frac{{\partial x}}{{\partial t}} + \frac{{\partial z}}{{\partial y}}\frac{{\partial y}}{{\partial t}} \cr & \frac{{\partial x}}{{\partial t}} = \frac{\partial }{{\partial t}}\left[ {s + t} \right] = 1 \cr & \frac{{\partial y}}{{\partial t}} = \frac{\partial }{{\partial t}}\left[ {st} \right] = s \cr & \frac{{\partial z}}{{\partial t}} = \frac{\partial }{{\partial t}}\left[ {s - t} \right] = - 1 \cr & {\text{Then}}{\text{,}} \cr & \frac{{\partial w}}{{\partial t}} = \left( {\frac{1}{{y + z}}} \right)\left( 1 \right) + \left( {\frac{{z - x}}{{{{\left( {y + z} \right)}^2}}}} \right)\left( s \right) + \left( { - \frac{{x + y}}{{{{\left( {y + z} \right)}^2}}}} \right)\left( { - 1} \right) \cr & \frac{{\partial w}}{{\partial t}} = \frac{1}{{y + z}} + \frac{{\left( {z - x} \right)s}}{{{{\left( {y + z} \right)}^2}}} + \frac{{x + y}}{{{{\left( {y + z} \right)}^2}}} \cr & \frac{{\partial w}}{{\partial t}} = \frac{{y + z + sz - sx + x + y}}{{{{\left( {y + z} \right)}^2}}} \cr & {\text{Write in terms of }}s{\text{ and }}t \cr & \frac{{\partial w}}{{\partial t}} = \frac{{st + s - t + s\left( {s - t} \right) - s\left( {s + t} \right) + s + t + st}}{{{{\left( {st + s - t} \right)}^2}}} \cr & \frac{{\partial w}}{{\partial t}} = \frac{{st + s - t + {s^2} - st - {s^2} - st + s + t + st}}{{{{\left( {st + s - t} \right)}^2}}} \cr & \frac{{\partial w}}{{\partial t}} = \frac{{2s}}{{{{\left( {st + s - t} \right)}^2}}} \cr} $$
