Chapter 12 - Functions of Several Veriables - 12.4 Partial Derivatives - 12.4 Exercises - Page 904: 36

$$\eqalign{ & {Q_{rr}}\left( {r,s} \right) = 0\,\,\,\,\,\,\,\,{\text{ }}{Q_{ss}}\left( {r,s} \right) = \frac{{2r}}{{{s^3}}} \cr & {Q_{rs}}\left( {r,s} \right) = {Q_{sr}}\left( {r,s} \right) = - \frac{1}{{{s^2}}} \cr} $$

$$\eqalign{ & Q\left( {r,s} \right) = r/s \cr & {\text{Find the first partial derivatives }}{Q_r}\left( {r,s} \right){\text{ and }}{Q_s}\left( {r,s} \right){\text{ then}} \cr & {Q_r}\left( {r,s} \right) = \frac{\partial }{{\partial r}}\left[ {\frac{r}{s}} \right] \cr & {\text{treat }}s{\text{ as a constant}}{\text{, then}} \cr & {Q_r}\left( {r,s} \right) = \frac{1}{s}\frac{\partial }{{\partial r}}\left[ r \right] \cr & {Q_r}\left( {r,s} \right) = \frac{1}{s} \cr & and \cr & {Q_s}\left( {r,s} \right) = \frac{\partial }{{\partial s}}\left[ {\frac{r}{s}} \right] \cr & {\text{treat }}r{\text{ as a constant}}{\text{, then}} \cr & {Q_s}\left( {r,s} \right) = r\frac{\partial }{{\partial s}}\left[ {\frac{1}{s}} \right] \cr & {Q_s}\left( {r,s} \right) = r\left( { - \frac{1}{{{s^2}}}} \right) \cr & {Q_s}\left( {r,s} \right) = - \frac{r}{{{s^2}}} \cr & \cr & {\text{Find the second partial derivatives }}{Q_{rs}}\left( {r,s} \right){\text{ and }}{Q_{sr}}\left( {r,s} \right){\text{ then}} \cr & {Q_{rs}}\left( {r,s} \right) = \frac{\partial }{{\partial s}}\left[ {\frac{1}{s}} \right] \cr & {\text{use product rule}} \cr & {Q_{rs}}\left( {r,s} \right) = - \frac{1}{{{s^2}}} \cr & and \cr & {\text{ }}{Q_{sr}}\left( {r,s} \right) = \frac{\partial }{{\partial r}}\left[ { - \frac{r}{{{s^2}}}} \right] \cr & {\text{ }}{Q_{sr}}\left( {r,s} \right) = - \frac{1}{{{s^2}}}\frac{\partial }{{\partial r}}\left[ r \right] \cr & {\text{ }}{Q_{sr}}\left( {r,s} \right) = - \frac{1}{{{s^2}}} \cr & \cr & {\text{Find the second partial derivatives }}{Q_{rr}}\left( {r,s} \right){\text{ and }}{Q_{ss}}\left( {r,s} \right){\text{ then}} \cr & {Q_{rr}}\left( {r,s} \right) = \frac{\partial }{{\partial r}}\left[ {\frac{1}{s}} \right] \cr & {Q_{rr}}\left( {r,s} \right) = 0 \cr & and \cr & {Q_{ss}}\left( {r,s} \right) = \frac{\partial }{{\partial s}}\left[ { - \frac{r}{{{s^2}}}} \right] \cr & {Q_{ss}}\left( {r,s} \right) = - r\frac{\partial }{{\partial s}}\left[ {{s^{ - 2}}} \right] \cr & {Q_{ss}}\left( {r,s} \right) = 2r{s^{ - 3}} \cr & {Q_{ss}}\left( {r,s} \right) = \frac{{2r}}{{{s^3}}} \cr} $$