Chapter 12 - Functions of Several Veriables - 12.5 The Chain Rule - 12.5 Exercises - Page 913: 21

$$\eqalign{ & {z_s} = 2s - 3{s^2} + {t^2} - 2st \cr & {z_t} = - 2t - {s^2} + 3{t^2} + 2st \cr} $$

$$\eqalign{ & {\text{Let the functions }}z = xy - {x^2}y,\,\,\,\,\,x = s + t{\text{ and }}y = s - t \cr & \cr & {\text{Calculate }}{z_s},{\text{ apply }}\frac{{\partial z}}{{\partial s}} = \frac{{\partial z}}{{\partial x}}\frac{{\partial x}}{{\partial s}} + \frac{{\partial z}}{{\partial y}}\frac{{\partial y}}{{\partial s}} \cr & \frac{{\partial z}}{{\partial x}} = \frac{\partial }{{\partial x}}\left[ {xy - {x^2}y} \right] = y - 2xy \cr & \frac{{\partial z}}{{\partial y}} = \frac{\partial }{{\partial y}}\left[ {xy - {x^2}y} \right] = x - {x^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[ {s - t} \right] = 1 \cr & {\text{Then}}{\text{,}} \cr & \frac{{\partial z}}{{\partial s}} = \left( {y - 2xy} \right)\left( 1 \right) + \left( {x - {x^2}} \right)\left( 1 \right) \cr & \frac{{\partial z}}{{\partial s}} = y - 2xy + x - {x^2} \cr & {\text{Write in terms of }}s{\text{ and }}t \cr & \frac{{\partial z}}{{\partial s}} = \left( {s - t} \right) - 2\left( {s + t} \right)\left( {s - t} \right) + \left( {s + t} \right) - {\left( {s + t} \right)^2} \cr & \frac{{\partial z}}{{\partial s}} = s - t - 2{s^2} + 2{t^2} + s + t - {s^2} - 2st - {t^2} \cr & \frac{{\partial z}}{{\partial s}} = 2s - 3{s^2} + {t^2} - 2st \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[ {s - t} \right] = - 1 \cr & {\text{Then}}{\text{,}} \cr & \frac{{\partial z}}{{\partial t}} = \left( {y - 2xy} \right)\left( 1 \right) + \left( {x - {x^2}} \right)\left( { - 1} \right) \cr & \frac{{\partial z}}{{\partial t}} = y - 2xy - x + {x^2} \cr & {\text{Write in terms of }}s{\text{ and }}t \cr & \frac{{\partial z}}{{\partial t}} = \left( {s - t} \right) - 2\left( {s + t} \right)\left( {s - t} \right) - \left( {s + t} \right) + {\left( {s + t} \right)^2} \cr & \frac{{\partial z}}{{\partial t}} = s - t - 2{s^2} + 2{t^2} - s - t + {s^2} + 2st + {t^2} \cr & \frac{{\partial z}}{{\partial t}} = - 2t - {s^2} + 3{t^2} + 2st \cr} $$