Chapter 13 - Parametric And Polar Curves; Conic Sections - 13.6 Directional Derivatives And Gradients - Exercises Set 13.6 - Page 968: 35

$$\nabla z = - \frac{{84}}{{{{\left( {6x - 7y} \right)}^2}}}\left( {y{\bf{i}} - x{\bf{j}}} \right)$$

$$\eqalign{ & z = \frac{{6x + 7y}}{{6x - 7y}} \cr & {\text{calculate the partial derivatives }}{z_x}{\text{ and }}{z_y} \cr & {z_x} = \frac{\partial }{{\partial x}}\left[ {\frac{{6x + 7y}}{{6x - 7y}}} \right] \cr & {\text{treat }}y{\text{ as a constant}}{\text{, and use the quotient rule for derivatives}} \cr & {z_x} = \frac{{\left( {6x - 7y} \right)\frac{\partial }{{\partial x}}\left[ {6x + 7y} \right] - \left( {6x + 7y} \right)\frac{\partial }{{\partial x}}\left[ {6x - 7y} \right]}}{{{{\left( {6x - 7y} \right)}^2}}} \cr & {z_x} = \frac{{\left( {6x - 7y} \right)\left( 6 \right) - \left( {6x + 7y} \right)\left( 6 \right)}}{{{{\left( {6x - 7y} \right)}^2}}} \cr & {z_x} = \frac{{36x - 42y - 36x - 42y}}{{{{\left( {6x - 7y} \right)}^2}}} \cr & {z_x} = \frac{{ - 84y}}{{{{\left( {6x - 7y} \right)}^2}}} \cr & and \cr & {z_y} = \frac{\partial }{{\partial y}}\left[ {\frac{{6x + 7y}}{{6x - 7y}}} \right] \cr & {\text{treat }}y{\text{ as a constant}}{\text{, and use the quotient rule for derivatives}} \cr & {z_y} = \frac{{\left( {6x - 7y} \right)\frac{\partial }{{\partial y}}\left[ {6x + 7y} \right] - \left( {6x + 7y} \right)\frac{\partial }{{\partial y}}\left[ {6x - 7y} \right]}}{{{{\left( {6x - 7y} \right)}^2}}} \cr & {z_y} = \frac{{\left( {6x - 7y} \right)\left( 7 \right) - \left( {6x + 7y} \right)\left( { - 7} \right)}}{{{{\left( {6x - 7y} \right)}^2}}} \cr & {z_y} = \frac{{42x - 49y + 42x + 49y}}{{{{\left( {6x - 7y} \right)}^2}}} \cr & {z_y} = \frac{{84x}}{{{{\left( {6x - 7y} \right)}^2}}} \cr & \cr & {\text{The gradient of the function }}z\left( {x,y} \right){\text{ is defined by }}\left( {{\text{see page 963}}} \right) \cr & \nabla z = {z_x}{\bf{i}} + {z_y}{\bf{j}} \cr & {\text{substituting the partial derivatives, we obtain}} \cr & \nabla z = \frac{{ - 84y}}{{{{\left( {6x - 7y} \right)}^2}}}{\bf{i}} + \frac{{84x}}{{{{\left( {6x - 7y} \right)}^2}}}{\bf{j}} \cr & {\text{factoring}} \cr & \nabla z = - \frac{{84}}{{{{\left( {6x - 7y} \right)}^2}}}\left( {y{\bf{i}} - x{\bf{j}}} \right) \cr} $$