Chapter 13 - Partial Derivatives - 13.3 Partial Derivatives - Exercises Set 13.3 - Page 937: 44

$$\eqalign{ & {f_x}\left( {x,y,z} \right) = {y^{ - 5/2}}z\sec \left( {\frac{{xz}}{y}} \right)\tan \left( {\frac{{xz}}{y}} \right) \cr & {f_y}\left( {x,y,z} \right) = - x{y^{ - 7/2}}z\sec \left( {\frac{{xz}}{y}} \right)\tan \left( {\frac{{xz}}{y}} \right) - \frac{3}{2}{y^{ - 5/2}}\sec \left( {\frac{{xz}}{y}} \right) \cr & {f_z}\left( {x,y,z} \right) = x{y^{ - 5/2}}\sec \left( {\frac{{xz}}{y}} \right)\tan \left( {\frac{{xz}}{y}} \right) \cr} $$

$$\eqalign{ & f\left( {x,y,z} \right) = {y^{ - 3/2}}\sec \left( {\frac{{xz}}{y}} \right) \cr & {\text{Calculate }}{f_x}\left( {x,y,z} \right) \cr & {f_x}\left( {x,y,z} \right) = \frac{\partial }{{\partial x}}\left[ {{y^{ - 3/2}}\sec \left( {\frac{{xz}}{y}} \right)} \right] \cr & {f_x}\left( {x,y,z} \right) = {y^{ - 3/2}}\frac{\partial }{{\partial x}}\left[ {\sec \left( {\frac{{xz}}{y}} \right)} \right] \cr & {f_x}\left( {x,y,z} \right) = {y^{ - 3/2}}\sec \left( {\frac{{xz}}{y}} \right)\tan \left( {\frac{{xz}}{y}} \right)\frac{\partial }{{\partial x}}\left[ {\frac{{xz}}{y}} \right] \cr & {f_x}\left( {x,y,z} \right) = {y^{ - 5/2}}z\sec \left( {\frac{{xz}}{y}} \right)\tan \left( {\frac{{xz}}{y}} \right) \cr & \cr & {\text{Calculate }}{f_y}\left( {x,y,z} \right) \cr & {f_y}\left( {x,y,z} \right) = \frac{\partial }{{\partial y}}\left[ {{y^{ - 3/2}}\sec \left( {\frac{{xz}}{y}} \right)} \right] \cr & {\text{Use the product rule}} \cr & {f_y}\left( {x,y,z} \right) = {y^{ - 3/2}}\frac{\partial }{{\partial y}}\left[ {\sec \left( {\frac{{xz}}{y}} \right)} \right] + \sec \left( {\frac{{xz}}{y}} \right)\frac{\partial }{{\partial y}}\left[ {{y^{ - 3/2}}} \right] \cr & {f_y}\left( {x,y,z} \right) = {y^{ - 3/2}}\sec \left( {\frac{{xz}}{y}} \right)\tan \left( {\frac{{xz}}{y}} \right)\left( { - \frac{{xz}}{{{y^2}}}} \right) + \sec \left( {\frac{{xz}}{y}} \right)\left( { - \frac{3}{2}{y^{ - 5/2}}} \right) \cr & {f_y}\left( {x,y,z} \right) = - x{y^{ - 7/2}}z\sec \left( {\frac{{xz}}{y}} \right)\tan \left( {\frac{{xz}}{y}} \right) - \frac{3}{2}{y^{ - 5/2}}\sec \left( {\frac{{xz}}{y}} \right) \cr & \cr & {\text{Calculate }}{f_z}\left( {x,y,z} \right) \cr & {f_z}\left( {x,y,z} \right) = \frac{\partial }{{\partial z}}\left[ {{y^{ - 3/2}}\sec \left( {\frac{{xz}}{y}} \right)} \right] \cr & {f_z}\left( {x,y,z} \right) = {y^{ - 3/2}}\frac{\partial }{{\partial z}}\left[ {\sec \left( {\frac{{xz}}{y}} \right)} \right] \cr & {f_z}\left( {x,y,z} \right) = {y^{ - 3/2}}\sec \left( {\frac{{xz}}{y}} \right)\tan \left( {\frac{{xz}}{y}} \right)\frac{\partial }{{\partial x}}\left[ {\frac{{xz}}{y}} \right] \cr & {f_z}\left( {x,y,z} \right) = {y^{ - 3/2}}\sec \left( {\frac{{xz}}{y}} \right)\tan \left( {\frac{{xz}}{y}} \right)\left( {\frac{x}{y}} \right) \cr & {f_z}\left( {x,y,z} \right) = x{y^{ - 5/2}}\sec \left( {\frac{{xz}}{y}} \right)\tan \left( {\frac{{xz}}{y}} \right) \cr} $$