Chapter 13 - Partial Derivatives - 13.4 Differentiability, Differentials, And Local Linearity - Exercises Set 13.4 - Page 947: 16

$$dw = yz{e^{xyz}}dx + xz{e^{xyz}}dy + xy{e^{xyz}}dz$$

$$\eqalign{ & w = {e^{xyz}} \cr & {\text{Let }}w = f\left( {x,y,z} \right){\text{ }} \cr & {\text{Calculate the partial derivative }}{f_x}\left( {x,y,z} \right){\text{, treating }}y{\text{ and }}z{\text{ as constants}} \cr & {\text{ }}{f_x}\left( {x,y,z} \right) = \frac{\partial }{{\partial x}}\left[ {{e^{xyz}}} \right] \cr & {\text{ }}{f_x}\left( {x,y,z} \right) = {e^{xyz}}\frac{\partial }{{\partial x}}\left[ {xyz} \right] \cr & {\text{ }}{f_x}\left( {x,y,z} \right) = {e^{xyz}}\left( {yz} \right) \cr & {\text{ }}{f_x}\left( {x,y,z} \right) = yz{e^{xyz}} \cr & {\text{Calculate the partial derivative }}{f_y}\left( {x,y,z} \right){\text{, treating }}x{\text{ and }}z{\text{ as constants}} \cr & {\text{ }}{f_y}\left( {x,y,z} \right) = \frac{\partial }{{\partial y}}\left[ {{e^{xyz}}} \right] \cr & {\text{ }}{f_y}\left( {x,y,z} \right) = {e^{xyz}}\frac{\partial }{{\partial y}}\left[ {xyz} \right] \cr & {\text{ }}{f_y}\left( {x,y,z} \right) = {e^{xyz}}\left( {xz} \right) \cr & {\text{ }}{f_y}\left( {x,y,z} \right) = xz{e^{xyz}} \cr & {\text{Calculate the partial derivative }}{f_z}\left( {x,y,z} \right){\text{, treating }}x{\text{ and }}y{\text{ as constants}} \cr & {\text{ }}{f_z}\left( {x,y,z} \right) = \frac{\partial }{{\partial z}}\left[ {{e^{xyz}}} \right] \cr & {\text{ }}{f_z}\left( {x,y,z} \right) = {e^{xyz}}\frac{\partial }{{\partial z}}\left[ {xyz} \right] \cr & {\text{ }}{f_z}\left( {x,y,z} \right) = {e^{xyz}}\left( {xy} \right) \cr & {\text{ }}{f_z}\left( {x,y,z} \right) = xy{e^{xyz}} \cr & \cr & {\text{The total differential of }}z{\text{ is given by}}:\,\,\left( {{\text{See page 944}}} \right) \cr & dw = {f_x}\left( {x,y,z} \right)dx + {f_y}\left( {x,y,z} \right)dy + {f_z}\left( {x,y,z} \right)dz \cr & {\text{Substitute the partial derivatives}} \cr & dw = yz{e^{xyz}}dx + xz{e^{xyz}}dy + xy{e^{xyz}}dz \cr} $$