Chapter 13 - Partial Derivatives - 13.6 Directional Derivatives And Gradients - Exercises Set 13.6 - Page 969: 46

$$54{\bf{i}} - 6{\bf{j}} + 9{\bf{k}}$$

$$\eqalign{ & f\left( {x,y,z} \right) = {y^2}z{\tan ^3}x;\,\,\,\,\,\left( {\pi /4, - 3,1} \right) \cr & \cr & {\text{calculate the partial derivatives }}{f_x}\left( {x,y,z} \right){\text{,}}\,{\text{ }}{f_y}\left( {x,y,z} \right)\,\,{\text{ and }}{f_z}\left( {x,y} \right) \cr & {f_x}\left( {x,y,z} \right) = \frac{\partial }{{\partial x}}\left[ {{y^2}z{{\tan }^3}x} \right] \cr & {\text{treat }}y{\text{ and }}z{\text{ as constants}} \cr & {f_x}\left( {x,y,z} \right) = {y^2}z\frac{\partial }{{\partial x}}\left[ {{{\tan }^3}x} \right] \cr & {f_x}\left( {x,y,z} \right) = {y^2}z\left( {3{{\tan }^2}x} \right)\left( {{{\sec }^2}x} \right) \cr & {f_x}\left( {x,y,z} \right) = 3z{\sec ^2}x{\tan ^2}x{y^2} \cr & \cr & {f_y}\left( {x,y,z} \right) = \frac{\partial }{{\partial y}}\left[ {{y^2}z{{\tan }^3}x} \right] \cr & {\text{treat }}x{\text{ and }}z{\text{ as a constant and use product rule}} \cr & {f_y}\left( {x,y,z} \right) = z{\tan ^3}x\frac{\partial }{{\partial y}}\left[ {{y^2}} \right] \cr & {f_y}\left( {x,y,z} \right) = 2yz{\tan ^3}x \cr & and \cr & {f_z}\left( {x,y,z} \right) = \frac{\partial }{{\partial z}}\left[ {{y^2}z{{\tan }^3}x} \right] \cr & {\text{treat }}x{\text{ and }}y{\text{ as constants}} \cr & {f_z}\left( {x,y,z} \right) = {y^2}{\tan ^3}x\frac{\partial }{{\partial z}}\left[ z \right] \cr & {f_z}\left( {x,y,z} \right) = {y^2}{\tan ^3}x \cr & \cr & {\text{The gradient of the function }}f\left( {x,y,z} \right){\text{ is defined by }}\left( {{\text{see page 963}}} \right) \cr & \nabla f\left( {x,y,z} \right) = {f_x}\left( {x,y,z} \right){\bf{i}} + {f_y}\left( {x,y,z} \right){\bf{j}} + {f_z}\left( {x,y,z} \right){\bf{k}} \cr & {\text{substituting the partial derivatives, we obtain}} \cr & \nabla f\left( {x,y,z} \right) = 3z{\sec ^2}x{\tan ^2}x{y^2}{\bf{i}} + 2yz{\tan ^3}x{\bf{j}} + {y^2}{\tan ^3}x{\bf{k}} \cr & \cr & {\text{the gradient of }}f{\text{ at }}\left( {\pi /4, - 3,1} \right){\text{ is}} \cr & \nabla f\left( {\pi /4, - 3,1} \right) = 3\left( 1 \right){\left( { - 3} \right)^2}{\sec ^2}\left( {\frac{\pi }{4}} \right){\tan ^2}\left( {\frac{\pi }{4}} \right){\bf{i}} + 2\left( { - 3} \right)\left( 1 \right){\tan ^3}\left( {\frac{\pi }{4}} \right){\bf{j}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + {\left( { - 3} \right)^2}{\tan ^3}\left( {\frac{\pi }{4}} \right){\bf{k}} \cr & \nabla f\left( {\pi /4, - 3,1} \right) =54{\bf{i}} - 6{\bf{j}} + 9{\bf{k}} \cr} $$