Chapter 13 - Partial Derivatives - 13.3 Partial Derivatives - Exercises Set 13.3 - Page 937: 49

$$\eqalign{ & \frac{{\partial w}}{{\partial x}} = \frac{x}{{\sqrt {{x^2} + {y^2} + {z^2}} }} \cr & \frac{{\partial w}}{{\partial y}} = \frac{y}{{\sqrt {{x^2} + {y^2} + {z^2}} }} \cr & \frac{{\partial w}}{{\partial z}} = \frac{z}{{\sqrt {{x^2} + {y^2} + {z^2}} }} \cr} $$

$$\eqalign{ & w = \sqrt {{x^2} + {y^2} + {z^2}} \cr & {\text{write as}} \cr & w = {\left( {{x^2} + {y^2} + {z^2}} \right)^{1/2}} \cr & \cr & {\text{Calculate }}\frac{{\partial w}}{{\partial x}} \cr & \frac{{\partial w}}{{\partial x}} = \frac{\partial }{{\partial x}}\left[ {{{\left( {{x^2} + {y^2} + {z^2}} \right)}^{1/2}}} \right] \cr & \frac{{\partial w}}{{\partial x}} = \frac{1}{2}{\left( {{x^2} + {y^2} + {z^2}} \right)^{ - 1/2}}\frac{\partial }{{\partial x}}\left[ {{x^2} + {y^2} + {z^2}} \right] \cr & \frac{{\partial w}}{{\partial x}} = \frac{1}{2}{\left( {{x^2} + {y^2} + {z^2}} \right)^{ - 1/2}}\left( {2x} \right) \cr & \frac{{\partial w}}{{\partial x}} = \frac{x}{{\sqrt {{x^2} + {y^2} + {z^2}} }} \cr & \cr & {\text{Calculate }}\frac{{\partial w}}{{\partial y}} \cr & \frac{{\partial w}}{{\partial y}} = \frac{\partial }{{\partial x}}\left[ {{{\left( {{x^2} + {y^2} + {z^2}} \right)}^{1/2}}} \right] \cr & \frac{{\partial w}}{{\partial y}} = \frac{1}{2}{\left( {{x^2} + {y^2} + {z^2}} \right)^{ - 1/2}}\frac{\partial }{{\partial y}}\left[ {{x^2} + {y^2} + {z^2}} \right] \cr & \frac{{\partial w}}{{\partial y}} = \frac{1}{2}{\left( {{x^2} + {y^2} + {z^2}} \right)^{ - 1/2}}\left( {2y} \right) \cr & \frac{{\partial w}}{{\partial y}} = \frac{y}{{\sqrt {{x^2} + {y^2} + {z^2}} }} \cr & \cr & {\text{Calculate }}\frac{{\partial w}}{{\partial z}} \cr & \frac{{\partial w}}{{\partial z}} = \frac{\partial }{{\partial z}}\left[ {{{\left( {{x^2} + {y^2} + {z^2}} \right)}^{1/2}}} \right] \cr & \frac{{\partial w}}{{\partial z}} = \frac{1}{2}{\left( {{x^2} + {y^2} + {z^2}} \right)^{ - 1/2}}\frac{\partial }{{\partial z}}\left[ {{x^2} + {y^2} + {z^2}} \right] \cr & \frac{{\partial w}}{{\partial z}} = \frac{1}{2}{\left( {{x^2} + {y^2} + {z^2}} \right)^{ - 1/2}}\left( {2z} \right) \cr & \frac{{\partial w}}{{\partial z}} = \frac{z}{{\sqrt {{x^2} + {y^2} + {z^2}} }} \cr} $$