Chapter 9 - Multiveriable Calculus - 9.2 Partial Derivatives - 9.2 Exercises - Page 478: 30

$$\eqalign{ & {k_{xx}} = - \frac{{25}}{{{{\left( {5x - 7y} \right)}^2}}}\,\,\,\,{k_{yy}} = \frac{{49}}{{{{\left( {5x - 7y} \right)}^2}}} \cr & {k_{xy}} = {k_{yx}} = \frac{{35}}{{{{\left( {5x - 7y} \right)}^2}}},\,\,\,\,\,\,{k_{yy}} = \frac{{49}}{{{{\left( {5x - 7y} \right)}^2}}} \cr} $$

$$\eqalign{ & k = \ln \left| {5x - 7y} \right| \cr & {\text{Find }}{k_x}{\text{ and }}{k_y} \cr & {k_x} = \frac{\partial }{{\partial x}}\left[ {\ln \left| {5x - 7y} \right|} \right] \cr & {\text{treat }}y{\text{ as a constant and use }}\left( {\ln u} \right)' = \frac{1}{u}\left( {u'} \right) \cr & {k_x} = \frac{1}{{5x - 7y}}\frac{\partial }{{\partial x}}\left[ {5x - 7y} \right] \cr & {\text{then}} \cr & {k_x} = \frac{1}{{5x - 7y}}\left( 5 \right) \cr & {k_x} = \frac{5}{{5x - 7y}} \cr & \cr & {k_y} = \frac{\partial }{{\partial y}}\left[ {\ln \left| {5x - 7y} \right|} \right] \cr & {\text{treat }}x{\text{ as a constant and use }}\left( {\ln u} \right)' = \frac{1}{u}\left( {u'} \right) \cr & {k_y} = \frac{1}{{5x - 7y}}\frac{\partial }{{\partial y}}\left[ {5x - 7y} \right] \cr & {\text{then}} \cr & {k_y} = \frac{1}{{5x - 7y}}\left( { - 7} \right) \cr & {k_y} = - \frac{7}{{5x - 7y}} \cr & \cr & {\text{find the second - order partial derivatives}} \cr & {k_{xy}} = \frac{\partial }{{\partial y}}\left[ {\frac{5}{{5x - 7y}}} \right] \cr & {k_{xy}} = 5\frac{\partial }{{\partial y}}\left[ {{{\left( {5x - 7y} \right)}^{ - 1}}} \right] \cr & {k_{xy}} = - 5{\left( {5x - 7y} \right)^{ - 2}}\frac{\partial }{{\partial y}}\left[ {5x - 7y} \right] \cr & {k_{xy}} = - 5{\left( {5x - 7y} \right)^{ - 2}}\left( { - 7} \right) \cr & {k_{xy}} = \frac{{35}}{{{{\left( {5x - 7y} \right)}^2}}} \cr & \cr & and \cr & {k_{yx}} = \frac{\partial }{{\partial x}}\left[ { - \frac{7}{{5x - 7y}}} \right] \cr & {k_{yx}} = - 7\frac{\partial }{{\partial x}}\left[ {{{\left( {5x - 7y} \right)}^{ - 1}}} \right] \cr & {k_{yx}} = 7{\left( {5x - 7y} \right)^{ - 2}}\frac{\partial }{{\partial x}}\left[ {5x - 7y} \right] \cr & {k_{yx}} = 7{\left( {5x - 7y} \right)^{ - 2}}\left( 5 \right) \cr & {k_{yx}} = \frac{{35}}{{{{\left( {5x - 7y} \right)}^2}}} \cr & \cr & {k_{xx}} = \frac{\partial }{{\partial x}}\left( {\frac{5}{{5x - 7y}}} \right) \cr & {k_{xx}} = 5\left( { - \frac{5}{{{{\left( {5x - 7y} \right)}^2}}}} \right) \cr & {k_{xx}} = - \frac{{25}}{{{{\left( {5x - 7y} \right)}^2}}} \cr & and \cr & {k_{yy}} = \frac{\partial }{{\partial y}}\left( { - \frac{7}{{5x - 7y}}} \right) \cr & {k_{yy}} = - 7\frac{\partial }{{\partial y}}\left( {\frac{1}{{5x - 7y}}} \right) \cr & {k_{yy}} = \frac{{49}}{{{{\left( {5x - 7y} \right)}^2}}} \cr} $$