Chapter 13 - Partial Derivatives - 13.4 Differentiability, Differentials, And Local Linearity - Exercises Set 13.4 - Page 947: 15

$$dw = 8dx - 3dy + 4dz$$

$$\eqalign{ & w = 8x - 3y + 4z \cr & \cr & {\text{Let }}w = f\left( {x,y,z} \right){\text{ }} \cr & {\text{Calculate the partial derivative }}{f_x}\left( {x,y,z} \right){\text{, treating }}y{\text{ and }}z{\text{ as constants}} \cr & {\text{ }}{f_x}\left( {x,y,z} \right) = \frac{\partial }{{\partial x}}\left[ {8x - 3y + 4z} \right] \cr & {\text{ }}{f_x}\left( {x,y,z} \right) = 8 - 0 + 0 \cr & {\text{ }}{f_x}\left( {x,y,z} \right) = 8 \cr & {\text{Calculate the partial derivative }}{f_y}\left( {x,y,z} \right){\text{, treating }}x{\text{ and }}z{\text{ as constants}} \cr & {\text{ }}{f_y}\left( {x,y,z} \right) = \frac{\partial }{{\partial y}}\left[ {8x - 3y + 4z} \right] \cr & {\text{ }}{f_y}\left( {x,y,z} \right) = 0 - 3 + 4 \cr & {\text{ }}{f_y}\left( {x,y,z} \right) = - 3 \cr & {\text{Calculate the partial derivative }}{f_z}\left( {x,y,z} \right){\text{, treating }}x{\text{ and }}y{\text{ as constants}} \cr & {\text{ }}{f_z}\left( {x,y,z} \right) = \frac{\partial }{{\partial z}}\left[ {8x - 3y + 4z} \right] \cr & {\text{ }}{f_z}\left( {x,y,z} \right) = 0 - 0 + 4 \cr & {\text{ }}{f_z}\left( {x,y,z} \right) = 4 \cr & \cr & {\text{The total differential of }}z{\text{ is given by}}:\,\,\left( {{\text{See page 944}}} \right) \cr & dw = {f_x}\left( {x,y,z} \right)dx + {f_y}\left( {x,y,z} \right)dy + {f_z}\left( {x,y,z} \right)dz \cr & {\text{Substitute the partial derivatives}} \cr & dw = 8dx - 3dy + 4dz \cr} $$