Chapter 13 - Parametric And Polar Curves; Conic Sections - 13.6 Directional Derivatives And Gradients - Exercises Set 13.6 - Page 968: 44

$$\nabla f\left( {3,4} \right) = - \left( {1/125} \right)\left( {3{\bf{i}} + 4{\bf{j}}} \right)$$

$$\eqalign{ & f\left( {x,y} \right) = {\left( {{x^2} + {y^2}} \right)^{ - 1/2}};\,\,\,\,\,\left( {3,4} \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} + {y^2}} \right)}^{ - 1/2}}} \right] \cr & {\text{treat }}y{\text{ as a constant and use the chain rule}} \cr & {f_x}\left( {x,y} \right) = \left( { - 1/2} \right){\left( {{x^2} + {y^2}} \right)^{ - 1/2 - 1}}\frac{\partial }{{\partial x}}\left( {{x^2} + {y^2}} \right) \cr & {f_x}\left( {x,y} \right) = \left( { - 1/2} \right){\left( {{x^2} + {y^2}} \right)^{ - 3/2}}\left( {2x} \right) \cr & {f_x}\left( {x,y} \right) = - x{\left( {{x^2} + {y^2}} \right)^{ - 3/2}} \cr & and \cr & {f_y}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {{{\left( {{x^2} + {y^2}} \right)}^{ - 1/2}}} \right] \cr & {\text{treat }}x{\text{ as a constant and use the chain rule}} \cr & {f_y}\left( {x,y} \right) = \left( { - 1/2} \right){\left( {{x^2} + {y^2}} \right)^{ - 1/2 - 1}}\frac{\partial }{{\partial y}}\left( {{x^2} + {y^2}} \right) \cr & {f_y}\left( {x,y} \right) = \left( { - 1/2} \right){\left( {{x^2} + {y^2}} \right)^{ - 3/2}}\left( {2y} \right) \cr & {f_y}\left( {x,y} \right) = - y{\left( {{x^2} + {y^2}} \right)^{ - 3/2}} \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) = - x{\left( {{x^2} + {y^2}} \right)^{ - 3/2}}{\bf{i}} - y{\left( {{x^2} + {y^2}} \right)^{ - 3/2}}{\bf{j}} \cr & \nabla f\left( {x,y} \right) = {\left( {{x^2} + {y^2}} \right)^{ - 3/2}}\left( { - x{\bf{i}} - y{\bf{j}}} \right) \cr & \cr & {\text{the gradient of }}f{\text{ at }}\left( {3,4} \right){\text{ is}} \cr & \nabla f\left( {3,4} \right) = {\left( {{3^2} + {4^2}} \right)^{ - 3/2}}\left( { - 3{\bf{i}} - 4{\bf{j}}} \right) \cr & \nabla f\left( {3,4} \right) = - {\left( {25} \right)^{ - 3/2}}\left( {3{\bf{i}} + 4{\bf{j}}} \right) \cr & \nabla f\left( {3,4} \right) = - \left( {1/125} \right)\left( {3{\bf{i}} + 4{\bf{j}}} \right) \cr} $$