Chapter 12 - Functions of Several Veriables - 12.4 Partial Derivatives - 12.4 Exercises - Page 904: 49

$$\eqalign{ & {F_u}\left( {u,v,w} \right) = \frac{1}{{v + w}} \cr & {F_v}\left( {u,v,w} \right) = {F_w}\left( {u,v,w} \right) = - \frac{u}{{{{\left( {v + w} \right)}^2}}} \cr} $$

$$\eqalign{ & F\left( {u,v,w} \right) = \frac{u}{{v + w}} \cr & \cr & {\text{Find the first partial derivative }}{F_u}\left( {u,v,w} \right) \cr & {F_u}\left( {u,v,w} \right) = \frac{\partial }{{\partial u}}\left[ {\frac{u}{{v + w}}} \right] \cr & {\text{treat }}v{\text{ and }}w{\text{ as a constant}}{\text{,}} \cr & {F_u}\left( {u,v,w} \right) = \frac{1}{{v + w}}\frac{\partial }{{\partial u}}\left[ u \right] \cr & {F_u}\left( {u,v,w} \right) = \frac{1}{{v + w}}\left( 1 \right) \cr & {F_u}\left( {u,v,w} \right) = \frac{1}{{v + w}} \cr & \cr & {\text{Find the first partial derivative }}{F_v}\left( {u,v,w} \right) \cr & {F_v}\left( {u,v,w} \right) = \frac{\partial }{{\partial v}}\left[ {\frac{u}{{v + w}}} \right] \cr & {\text{treat }}u{\text{ and }}w{\text{ as a constant}}{\text{,}} \cr & {F_v}\left( {u,v,w} \right) = u\frac{\partial }{{\partial v}}\left[ {{{\left( {v + w} \right)}^{ - 1}}} \right] \cr & {\text{chain rule}} \cr & {F_v}\left( {u,v,w} \right) = - u{\left( {v + w} \right)^{ - 2}}\frac{\partial }{{\partial v}}\left[ {v + w} \right] \cr & {F_v}\left( {u,v,w} \right) = - u{\left( {v + w} \right)^{ - 2}} \cr & {F_v}\left( {u,v,w} \right) = - \frac{u}{{{{\left( {v + w} \right)}^2}}} \cr & \cr & {\text{Find the first partial derivative }}{F_w}\left( {u,v,w} \right) \cr & {F_w}\left( {u,v,w} \right) = \frac{\partial }{{\partial w}}\left[ {\frac{u}{{v + w}}} \right] \cr & {\text{treat }}u{\text{ and }}v{\text{ as a constant}}{\text{,}} \cr & {F_w}\left( {u,v,w} \right) = u\frac{\partial }{{\partial w}}\left[ {{{\left( {v + w} \right)}^{ - 1}}} \right] \cr & {\text{chain rule}} \cr & {F_w}\left( {u,v,w} \right) = - u{\left( {v + w} \right)^{ - 2}}\frac{\partial }{{\partial w}}\left[ {v + w} \right] \cr & {F_w}\left( {u,v,w} \right) = - u{\left( {v + w} \right)^{ - 2}} \cr & {F_w}\left( {u,v,w} \right) = - \frac{u}{{{{\left( {v + w} \right)}^2}}} \cr} $$