Chapter 13 - Partial Derivatives - 13.4 Differentiability, Differentials, And Local Linearity - Exercises Set 13.4 - Page 947: 20

$$dw = \frac{1}{{2\sqrt x }}dx + \frac{1}{{2\sqrt y }}dy + \frac{1}{{2\sqrt z }}dz$$

$$\eqalign{ & w = \sqrt x + \sqrt y + \sqrt z \cr & {\text{Let }}w = f\left( {x,y,z} \right) = {x^{1/2}} + {y^{1/2}} + {z^{1/2}}{\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[ {{x^{1/2}} + {y^{1/2}} + {z^{1/2}}{\text{ }}} \right] \cr & {\text{ }}{f_x}\left( {x,y,z} \right) = \frac{\partial }{{\partial x}}\left[ {{x^{1/2}}{\text{ }}} \right] + \frac{\partial }{{\partial x}}\left[ {{y^{1/2}}{\text{ }}} \right] + \frac{\partial }{{\partial x}}\left[ {{z^{1/2}}{\text{ }}} \right] \cr & {\text{ }}{f_x}\left( {x,y,z} \right) = \frac{1}{2}{x^{ - 1/2}} \cr & {\text{ }}{f_x}\left( {x,y,z} \right) = \frac{1}{{2\sqrt x }} \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[ {{x^{1/2}} + {y^{1/2}} + {z^{1/2}}{\text{ }}} \right] \cr & {\text{ }}{f_y}\left( {x,y,z} \right) = \frac{\partial }{{\partial y}}\left[ {{x^{1/2}}{\text{ }}} \right] + \frac{\partial }{{\partial y}}\left[ {{y^{1/2}}{\text{ }}} \right] + \frac{\partial }{{\partial y}}\left[ {{z^{1/2}}{\text{ }}} \right] \cr & {\text{ }}{f_y}\left( {x,y,z} \right) = \frac{1}{2}{y^{ - 1/2}} \cr & {\text{ }}{f_y}\left( {x,y,z} \right) = \frac{1}{{2\sqrt y }} \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[ {{x^{1/2}} + {y^{1/2}} + {z^{1/2}}{\text{ }}} \right] \cr & {\text{ }}{f_z}\left( {x,y,z} \right) = \frac{\partial }{{\partial z}}\left[ {{x^{1/2}}{\text{ }}} \right] + \frac{\partial }{{\partial z}}\left[ {{y^{1/2}}{\text{ }}} \right] + \frac{\partial }{{\partial z}}\left[ {{z^{1/2}}{\text{ }}} \right] \cr & {\text{ }}{f_z}\left( {x,y,z} \right) = \frac{1}{2}{z^{ - 1/2}} \cr & {\text{ }}{f_z}\left( {x,y,z} \right) = \frac{1}{{2\sqrt z }} \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 = \frac{1}{{2\sqrt x }}dx + \frac{1}{{2\sqrt y }}dy + \frac{1}{{2\sqrt z }}dz \cr} $$