Chapter 12 - Functions of Several Veriables - Review Exercises - Page 961: 52

$${w_s} = 8{s^3}{t^4}{\text{ and }}{w_t} = 8{s^4}{t^3}$$

$$\eqalign{ & w = xyz,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} x = 2st,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} y = s{t^2},{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} z = {s^2}t \cr & {\text{Calculate the partial derivatives }}\frac{{\partial w}}{{\partial x}},{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \frac{{\partial w}}{{\partial y}},{\mkern 1mu} {\mkern 1mu} \frac{{\partial w}}{{\partial z}} \cr & \frac{{\partial w}}{{\partial x}} = yz \cr & \frac{{\partial w}}{{\partial y}} = xz \cr & \frac{{\partial w}}{{\partial z}} = xy \cr & {\text{Calculate the derivatives }}\frac{{\partial x}}{{\partial s}},{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \frac{{\partial y}}{{\partial s}},{\mkern 1mu} {\mkern 1mu} \frac{{\partial z}}{{\partial s}}{\text{ and }}\frac{{\partial x}}{{\partial t}},{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \frac{{\partial y}}{{\partial t}},{\mkern 1mu} {\mkern 1mu} \frac{{\partial z}}{{\partial t}} \cr & \frac{{\partial x}}{{\partial s}} = 2t,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \frac{{\partial y}}{{\partial s}} = {t^2},{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \frac{{\partial z}}{{\partial s}} = 2st \cr & \frac{{\partial x}}{{\partial t}} = 2s,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \frac{{\partial y}}{{\partial t}} = 2st,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \frac{{\partial z}}{{\partial t}} = {s^2} \cr & {\text{Use the chain rule }}\frac{{\partial w}}{{\partial s}} = \frac{{\partial w}}{{\partial x}}\frac{{\partial x}}{{\partial s}} + {\mkern 1mu} {\mkern 1mu} \frac{{\partial w}}{{\partial y}}\frac{{\partial y}}{{\partial s}} + {\mkern 1mu} {\mkern 1mu} \frac{{\partial w}}{{\partial z}}\frac{{\partial z}}{{\partial s}} \cr & \frac{{\partial w}}{{\partial s}} = \left( {yz} \right)\left( {2t} \right) + {\mkern 1mu} \left( {xz} \right)\left( {{t^2}} \right) + {\mkern 1mu} \left( {xy} \right)\left( {2st} \right) \cr & \frac{{\partial w}}{{\partial s}} = 2tyz + {\mkern 1mu} xz{t^2} + 2xyst \cr & {\text{Write in terms of }}s \cr & \frac{{\partial w}}{{\partial s}} = 2t\left( {s{t^2}} \right)\left( {{s^2}t} \right) + {\mkern 1mu} \left( {2st} \right)\left( {{s^2}t} \right){t^2} + 2\left( {2st} \right)\left( {s{t^2}} \right)st \cr & \frac{{\partial w}}{{\partial s}} = 2{s^3}{t^4} + {\mkern 1mu} 2{s^3}{t^4} + 4{s^3}{t^4}{\mkern 1mu} \cr & \frac{{\partial w}}{{\partial s}} = 8{s^3}{t^4} \cr & or \cr & {w_s} = 8{s^3}{t^4} \cr & \cr & {\text{Use the chain rule }}\frac{{\partial w}}{{\partial t}} = \frac{{\partial w}}{{\partial x}}\frac{{\partial x}}{{\partial t}} + {\mkern 1mu} {\mkern 1mu} \frac{{\partial w}}{{\partial y}}\frac{{\partial y}}{{\partial t}} + {\mkern 1mu} {\mkern 1mu} \frac{{\partial w}}{{\partial z}}\frac{{\partial z}}{{\partial t}} \cr & \frac{{\partial w}}{{\partial t}} = \left( {yz} \right)\left( {2s} \right) + {\mkern 1mu} \left( {xz} \right)\left( {2st} \right) + \left( {xy} \right)\left( {{s^2}} \right) \cr & \frac{{\partial w}}{{\partial t}} = 2syz + 2xzst + xy{s^2} \cr & {\text{Write in terms of }}t \cr & \frac{{\partial w}}{{\partial t}} = 2s\left( {s{t^2}} \right)\left( {{s^2}t} \right) + 2\left( {2st} \right)\left( {{s^2}t} \right)st + \left( {2st} \right)\left( {s{t^2}} \right){s^2} \cr & \frac{{\partial w}}{{\partial t}} = 2{s^4}{t^3} + 4{s^4}{t^3} + 2{s^4}{t^3} \cr & \frac{{\partial w}}{{\partial t}} = 8{s^4}{t^3} \cr & or \cr & {w_t} = 8{s^4}{t^3} \cr} $$