Chapter 13 - Partial Derivatives - 13.5 The Chain Rule - Exercises Set 13.5 - Page 957: 32

$${\left. {\,\,\,\,\,\,\,{{\left. {\frac{{\partial f}}{{\partial u}}} \right|}_{u = 1,\,\,v = - 2}} = \frac{{351}}{2}{\text{ and }}\frac{{\partial f}}{{\partial v}}} \right|_{u = 1,\,\,v = - 2}} = - 168$$

$$\eqalign{ & {\text{Let }}f\left( {x,y} \right) = {x^2}{y^2} - x + 2y,\,\,\,\,\,\,\,x = \sqrt u ,\,\,\,\,\,\,\,\,y = u{v^3} \cr & \cr & {\text{Calculate the partial derivatives }}\frac{{\partial f}}{{\partial x}}{\text{ and }}\frac{{\partial f}}{{\partial y}} \cr & \,\,\,\,\,\,\frac{{\partial f}}{{\partial x}} = \frac{\partial }{{\partial x}}\left[ {{x^2}{y^2} - x + 2y} \right] \cr & \,\,\,\,\,\,\frac{{\partial f}}{{\partial x}} = 2x{y^2} - 1 \cr & \,\,\,\,\,\,\frac{{\partial f}}{{\partial y}} = \frac{\partial }{{\partial y}}\left[ {{x^2}{y^2} - x + 2y} \right] \cr & \,\,\,\,\,\,\frac{{\partial f}}{{\partial y}} = 2{x^2}y + 2 \cr & {\text{Calculate the partial derivatives }}\frac{{\partial x}}{{\partial u}}{\text{ and }}\frac{{\partial y}}{{\partial u}} \cr & \,\,\,\,\,\,\frac{{\partial x}}{{\partial u}} = \frac{\partial }{{\partial u}}\left[ {\sqrt u } \right] = \frac{1}{{2\sqrt u }} \cr & \,\,\,\,\,\,\frac{{\partial y}}{{\partial u}} = \frac{\partial }{{\partial u}}\left[ {u{v^3}} \right] = {v^3} \cr & {\text{Calculate the partial derivatives }}\frac{{\partial x}}{{\partial v}}{\text{ and }}\frac{{\partial y}}{{\partial v}} \cr & \,\,\,\,\,\,\frac{{\partial x}}{{\partial v}} = \frac{\partial }{{\partial v}}\left[ {\sqrt u } \right] = 0 \cr & \,\,\,\,\,\,\frac{{\partial y}}{{\partial v}} = \frac{\partial }{{\partial v}}\left[ {u{v^3}} \right] = 3{v^2} \cr & \cr & {\text{Calculate the partial derivatives }}\frac{{\partial f}}{{\partial u}}{\text{ and }}\frac{{\partial f}}{{\partial u}} \cr & \,\,\,\,\,\,\,\frac{{\partial f}}{{\partial u}} = \frac{{\partial f}}{{\partial x}}\frac{{\partial x}}{{\partial u}} + \frac{{\partial f}}{{\partial y}}\frac{{\partial y}}{{\partial u}} \cr & \,\,\,\,\,\,\,\frac{{\partial f}}{{\partial u}} = \left( {2x{y^2} - 1} \right)\left( {\frac{1}{{2\sqrt u }}} \right) + \left( {2{x^2}y + 2} \right)\left( {{v^3}} \right) \cr & \,\,\,\,\,\,\,\frac{{\partial f}}{{\partial u}} = \left( {2\sqrt u {{\left( {u{v^3}} \right)}^2} - 1} \right)\left( {\frac{1}{{2\sqrt u }}} \right) + \left( {2{{\left( {\sqrt u } \right)}^2}u{v^3} + 2} \right)\left( {{v^3}} \right) \cr & \,\,\,\,\,\,\,{\left. {\frac{{\partial f}}{{\partial u}}} \right|_{u = 1,\,\,v = - 2}} \cr & \,\,\,\,\,\,\,{\left. {\frac{{\partial f}}{{\partial u}}} \right|_{u = 1,\,\,v = - 2}} = \left( {2\sqrt 1 {{\left( {\left( 1 \right){{\left( { - 2} \right)}^3}} \right)}^2} - 1} \right)\left( {\frac{1}{{2\sqrt 1 }}} \right) + \left( {2{{\left( {\sqrt 1 } \right)}^2}\left( 1 \right){{\left( { - 2} \right)}^3} + 2} \right)\left( {{{\left( { - 2} \right)}^3}} \right) \cr & \,\,\,\,\,\,\,{\left. {\frac{{\partial f}}{{\partial u}}} \right|_{u = 1,\,\,v = - 2}} = \frac{{127}}{2} + 112 \cr & \,\,\,\,\,\,\,{\left. {\frac{{\partial f}}{{\partial u}}} \right|_{u = 1,\,\,v = - 2}} = \frac{{351}}{2} \cr & \cr & \,\,\,\,\,\,\,\frac{{\partial f}}{{\partial v}} = \frac{{\partial f}}{{\partial x}}\frac{{\partial x}}{{\partial v}} + \frac{{\partial f}}{{\partial y}}\frac{{\partial y}}{{\partial v}} \cr & \,\,\,\,\,\,\,\frac{{\partial f}}{{\partial v}} = \left( {2x{y^2} - 1} \right)\left( 0 \right) + \left( {2{x^2}y + 2} \right)\left( {3{v^2}} \right) \cr & \,\,\,\,\,\,\,\frac{{\partial f}}{{\partial v}} = \left( {2{x^2}y + 2} \right)\left( {3{v^2}} \right) \cr & \,\,\,\,\,\,\,{\left. {\frac{{\partial f}}{{\partial v}}} \right|_{u = 1,\,\,v = - 2}} = \left( {2{{\left( {\sqrt u } \right)}^2}u{v^3} + 2} \right)\left( {3{v^2}} \right) \cr & \,\,\,\,\,\,\,{\left. {\frac{{\partial f}}{{\partial v}}} \right|_{u = 1,\,\,v = - 2}} = \left( {2{{\left( {\sqrt 1 } \right)}^2}\left( 1 \right){{\left( { - 2} \right)}^3} + 2} \right)\left( {3{{\left( { - 2} \right)}^2}} \right) \cr & \,\,\,\,\,\,\,{\left. {\frac{{\partial f}}{{\partial v}}} \right|_{u = 1,\,\,v = - 2}} = - 168 \cr} $$