Chapter 13 - Functions of Several Variables - Review Exercises - Page 961: 39

$$\frac{{\partial w}}{{\partial r}} = \frac{{4{r^2}t - 4r{t^2} - {t^3}}}{{{{\left( {2r - t} \right)}^2}}}{\text{ and }}\frac{{\partial w}}{{\partial t}} = \frac{{4{r^2}t - r{t^2} + 4{r^3}}}{{{{\left( {2r - t} \right)}^2}}}$$

$$\eqalign{ & w = \frac{{xy}}{z},{\text{ }}x = 2r + t,{\text{ }}y = rt,{\text{ }}z = 2r - t \cr & \left( {\text{a}} \right){\text{ Calculate }}\frac{{\partial w}}{{\partial r}}{\text{ by using the chain rule}} \cr & \frac{{\partial w}}{{\partial r}} = \frac{{\partial w}}{{\partial x}}\frac{{\partial x}}{{\partial r}} + \frac{{\partial w}}{{\partial y}}\frac{{\partial y}}{{\partial r}} + \frac{{\partial w}}{{\partial z}}\frac{{\partial z}}{{\partial r}} \cr & \frac{{\partial w}}{{\partial r}} = \frac{\partial }{{\partial x}}\left[ {\frac{{xy}}{z}} \right]\frac{\partial }{{\partial r}}\left[ {2r + t} \right] + \frac{\partial }{{\partial y}}\left[ {\frac{{xy}}{z}} \right]\frac{\partial }{{\partial r}}\left[ {rt} \right] \cr & {\text{ }} + \frac{\partial }{{\partial z}}\left[ {\frac{{xy}}{z}} \right]\frac{\partial }{{\partial r}}\left[ {2r - t} \right] \cr & {\text{Computing derivatives}} \cr & \frac{{\partial w}}{{\partial r}} = \left( {\frac{y}{z}} \right)\left( 2 \right) + \left( {\frac{x}{z}} \right)\left( t \right) + \left( { - \frac{{xy}}{{{z^2}}}} \right)\left( 2 \right) \cr & {\text{Simplifying}} \cr & \frac{{\partial w}}{{\partial r}} = \frac{{2y}}{z} + \frac{{xt}}{z} - \frac{{2xy}}{{{z^2}}} \cr & \frac{{\partial w}}{{\partial r}} = \frac{{2yz + xtz - 2xy}}{{{z^2}}} \cr & {\text{Write in terms of }}t,r \cr & \frac{{\partial w}}{{\partial r}} = \frac{{2\left( {rt} \right)\left( {2r - t} \right) + \left( {2r + t} \right)\left( {2r - t} \right)t - 2rt\left( {2r + t} \right)}}{{{{\left( {2r - t} \right)}^2}}} \cr & \frac{{\partial w}}{{\partial r}} = \frac{{4{r^2}t - 2r{t^2} + 4{r^2}t - {t^3} - 4{r^2}t - 2r{t^2}}}{{{{\left( {2r - t} \right)}^2}}} \cr & \frac{{\partial w}}{{\partial r}} = \frac{{4{r^2}t - 4r{t^2} - {t^3}}}{{{{\left( {2r - t} \right)}^2}}} \cr & \cr & {\text{Calculate }}\frac{{\partial w}}{{\partial t}}{\text{ by using the chain rule}} \cr & \frac{{\partial w}}{{\partial t}} = \frac{{\partial w}}{{\partial x}}\frac{{\partial x}}{{\partial t}} + \frac{{\partial w}}{{\partial y}}\frac{{\partial y}}{{\partial t}} + \frac{{\partial w}}{{\partial z}}\frac{{\partial z}}{{\partial t}} \cr & \frac{{\partial w}}{{\partial t}} = \frac{\partial }{{\partial x}}\left[ {\frac{{xy}}{z}} \right]\frac{\partial }{{\partial t}}\left[ {2r + t} \right] + \frac{\partial }{{\partial y}}\left[ {\frac{{xy}}{z}} \right]\frac{\partial }{{\partial t}}\left[ {rt} \right] \cr & {\text{ }} + \frac{\partial }{{\partial z}}\left[ {\frac{{xy}}{z}} \right]\frac{\partial }{{\partial t}}\left[ {2r - t} \right] \cr & {\text{Computing derivatives}} \cr & \frac{{\partial w}}{{\partial r}} = \left( {\frac{y}{z}} \right)\left( 1 \right) + \left( {\frac{x}{z}} \right)\left( r \right) + \left( { - \frac{{xy}}{{{z^2}}}} \right)\left( { - 1} \right) \cr & \frac{{\partial w}}{{\partial r}} = \frac{y}{z} + \frac{{xr}}{z} + \frac{{xy}}{{{z^2}}} \cr & \frac{{\partial w}}{{\partial r}} = \frac{{yz + xrz + xy}}{{{z^2}}} \cr & {\text{Write in terms of }}t,r \cr & \frac{{\partial w}}{{\partial r}} = \frac{{rt\left( {2r - t} \right) + \left( {2r + t} \right)r\left( {2r - t} \right) + \left( {2r + t} \right)rt}}{{{{\left( {2r - t} \right)}^2}}} \cr & \frac{{\partial w}}{{\partial r}} = \frac{{2{r^2}t - r{t^2} + r\left( {4{r^2} - {t^2}} \right) + 2{r^2}t + r{t^2}}}{{{{\left( {2r - t} \right)}^2}}} \cr & \frac{{\partial w}}{{\partial r}} = \frac{{4{r^2}t + 4{r^3} - r{t^2}}}{{{{\left( {2r - t} \right)}^2}}} \cr & \cr & \left( {\text{b}} \right){\text{By converting }}w{\text{ to a function of }}r{\text{,}}t{\text{ before differentiating}} \cr & w = \frac{{xy}}{z},{\text{ }}x = 2r + t,{\text{ }}y = rt,{\text{ }}z = 2r - t \cr & w = \frac{{\left( {2r + t} \right)rt}}{{2r - t}} \cr & w = \frac{{2{r^2}t + {t^2}r}}{{2r - t}} \cr & \frac{{\partial w}}{{\partial t}} = \frac{\partial }{{\partial t}}\left[ {\frac{{2{r^2}t + {t^2}r}}{{2r - t}}} \right] \cr & \frac{{\partial w}}{{\partial t}} = \frac{{\left( {2r - t} \right)\left( {2{r^2} + {t^2}} \right) - \left( {2{r^2}t + {t^2}r} \right)\left( { - 1} \right)}}{{{{\left( {2r - t} \right)}^2}}} \cr & \frac{{\partial w}}{{\partial t}} = \frac{{4{r^3} + 2rt - 2t{r^2} + 2{r^2}t + {t^2}r}}{{{{\left( {2r - t} \right)}^2}}} \cr & \frac{{\partial w}}{{\partial t}} = \frac{{4{r^2}t - r{t^2} + 4{r^3}}}{{{{\left( {2r - t} \right)}^2}}} \cr & \cr & \frac{{\partial w}}{{\partial r}} = \frac{\partial }{{\partial r}}\left[ {\frac{{2{r^2}t + {t^2}r}}{{2r - t}}} \right] \cr & \frac{{\partial w}}{{\partial r}} = \frac{{\left( {2r - t} \right)\left( {4rt + {t^2}} \right) - \left( {2{r^2}t + {t^2}r} \right)\left( 2 \right)}}{{{{\left( {2r - t} \right)}^2}}} \cr & \frac{{\partial w}}{{\partial r}} = \frac{{8{r^2}t + 2r{t^2} - 4r{t^2} - 4{r^2}t - 2r{t^2}}}{{{{\left( {2r - t} \right)}^2}}} \cr & \frac{{\partial w}}{{\partial r}} = \frac{{4{r^2}t - 4r{t^2} - {t^3}}}{{{{\left( {2r - t} \right)}^2}}} \cr} $$