Chapter 12 - Functions of Several Veriables - 12.4 Partial Derivatives - 12.4 Exercises - Page 904: 50

$$\eqalign{ & {G_r}\left( {r,s,t} \right) = \frac{{s + t}}{{2\sqrt {rs + rt + st} }} \cr & {G_s}\left( {r,s,t} \right) = \frac{{r + t}}{{2\sqrt {rs + rt + st} }} \cr & {G_t}\left( {r,s,t} \right) = \frac{{r + s}}{{2\sqrt {rs + rt + st} }} \cr} $$

$$\eqalign{ & G\left( {r,s,t} \right) = \sqrt {rs + rt + st} \cr & G\left( {r,s,t} \right) = {\left( {rs + rt + st} \right)^{1/2}} \cr & {\text{Find the first partial derivative }}{G_r}\left( {r,s,t} \right) \cr & {G_r}\left( {r,s,t} \right) = \frac{\partial }{{\partial r}}\left[ {{{\left( {rs + rt + st} \right)}^{1/2}}} \right] \cr & {\text{treat }}s{\text{ and }}t{\text{ as a constants}}{\text{, then use chain rule}} \cr & {G_r}\left( {r,s,t} \right) = \frac{1}{2}{\left( {rs + rt + st} \right)^{ - 1/2}}\frac{\partial }{{\partial r}}\left[ {rs + rt + st} \right] \cr & {G_r}\left( {r,s,t} \right) = \frac{1}{2}{\left( {rs + rt + st} \right)^{ - 1/2}}\left( {s + t} \right) \cr & {\text{simplifying}} \cr & {G_r}\left( {r,s,t} \right) = \frac{{s + t}}{{2\sqrt {rs + rt + st} }} \cr & \cr & {\text{Find the first partial derivative }}{G_s}\left( {r,s,t} \right) \cr & {G_s}\left( {r,s,t} \right) = \frac{\partial }{{\partial s}}\left[ {{{\left( {rs + rt + st} \right)}^{1/2}}} \right] \cr & {\text{treat }}s{\text{ and }}t{\text{ as a constants}}{\text{, then use chain rule}} \cr & {G_s}\left( {r,s,t} \right) = \frac{1}{2}{\left( {rs + rt + st} \right)^{ - 1/2}}\frac{\partial }{{\partial s}}\left[ {rs + rt + st} \right] \cr & {G_s}\left( {r,s,t} \right) = \frac{1}{2}{\left( {rs + rt + st} \right)^{ - 1/2}}\left( {r + t} \right) \cr & {\text{simplifying}} \cr & {G_s}\left( {r,s,t} \right) = \frac{{r + t}}{{2\sqrt {rs + rt + st} }} \cr & \cr & {\text{Find the first partial derivative }}{G_t}\left( {r,s,t} \right) \cr & {G_t}\left( {r,s,t} \right) = \frac{\partial }{{\partial t}}\left[ {{{\left( {rs + rt + st} \right)}^{1/2}}} \right] \cr & {\text{treat }}r{\text{ and }}s{\text{ as a constants}}{\text{, then use chain rule}} \cr & {G_t}\left( {r,s,t} \right) = \frac{1}{2}{\left( {rs + rt + st} \right)^{ - 1/2}}\frac{\partial }{{\partial t}}\left[ {rs + rt + st} \right] \cr & {G_t}\left( {r,s,t} \right) = \frac{1}{2}{\left( {rs + rt + st} \right)^{ - 1/2}}\left( {r + s} \right) \cr & {\text{simplifying}} \cr & {G_t}\left( {r,s,t} \right) = \frac{{r + s}}{{2\sqrt {rs + rt + st} }} \cr} $$