Chapter 13 - Parametric And Polar Curves; Conic Sections - 13.6 Directional Derivatives And Gradients - Exercises Set 13.6 - Page 968: 43

$$\nabla f\left( { - 1, - 1} \right) = - 12\left( {3{\bf{i}} + {\bf{j}}} \right)$$

$$\eqalign{ & f\left( {x,y} \right) = {\left( {{x^2} + xy} \right)^3};\,\,\,\,\,\left( { - 1, - 1} \right) \cr & \cr & {\text{calculate the partial derivatives }}{f_x}\left( {x,y} \right){\text{ and }}{f_y}\left( {x,y} \right) \cr & {f_x}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {{{\left( {{x^2} + xy} \right)}^3}} \right] \cr & {\text{treat }}y{\text{ as a constant and use the chain rule}} \cr & {f_x}\left( {x,y} \right) = 3{\left( {{x^2} + xy} \right)^{3 - 1}}\frac{\partial }{{\partial x}}\left( {{x^2} + xy} \right) \cr & {f_x}\left( {x,y} \right) = 3{\left( {{x^2} + xy} \right)^2}\left( {2x + y} \right) \cr & {f_x}\left( {x,y} \right) = 3\left( {2x + y} \right){\left( {{x^2} + xy} \right)^2} \cr & and \cr & {f_y}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {5{x^2} + {y^4}} \right] \cr & {\text{treat }}x{\text{ as a constant and use the chain rule}} \cr & {f_y}\left( {x,y} \right) = 3{\left( {{x^2} + xy} \right)^{3 - 1}}\frac{\partial }{{\partial y}}\left( {{x^2} + xy} \right) \cr & {f_y}\left( {x,y} \right) = 3{\left( {{x^2} + xy} \right)^2}\left( {0 + x} \right) \cr & {f_y}\left( {x,y} \right) = 3x{\left( {{x^2} + xy} \right)^2} \cr & \cr & {\text{The gradient of the function }}f\left( {x,y} \right){\text{ is defined by }}\left( {{\text{see page 963}}} \right) \cr & \nabla f\left( {x,y} \right) = {f_x}\left( {x,y} \right){\bf{i}} + {f_y}\left( {x,y} \right){\bf{j}} \cr & {\text{substituting the partial derivatives, we obtain}} \cr & \nabla f\left( {x,y} \right) = 3\left( {2x + y} \right){\left( {{x^2} + xy} \right)^2}{\bf{i}} + 3x{\left( {{x^2} + xy} \right)^2}{\bf{j}} \cr & \nabla f\left( {x,y} \right) = 3{\left( {{x^2} + xy} \right)^2}\left[ {\left( {2x + y} \right){\bf{i}} + x{\bf{j}}} \right] \cr & \cr & {\text{the gradient of }}f{\text{ at }}\left( { - 1, - 1} \right){\text{ is}} \cr & \nabla f\left( { - 1, - 1} \right) = 3{\left( {{{\left( { - 1} \right)}^2} + \left( { - 1} \right)\left( { - 1} \right)} \right)^2}\left[ {\left( {2\left( { - 1} \right) - 1} \right){\bf{i}} + \left( { - 1} \right){\bf{j}}} \right] \cr & \nabla f\left( { - 1, - 1} \right) = 3{\left( 2 \right)^2}\left[ {\left( { - 3} \right){\bf{i}} + \left( { - 1} \right){\bf{j}}} \right] \cr & \nabla f\left( { - 1, - 1} \right) = - 12\left( {3{\bf{i}} + {\bf{j}}} \right) \cr} $$