Chapter 13 - Parametric And Polar Curves; Conic Sections - 13.6 Directional Derivatives And Gradients - Exercises Set 13.6 - Page 968: 2

$${D_{\bf{u}}}f\left( {3,1} \right) = \frac{{27}}{5}$$

$$\eqalign{ & f\left( {x,y} \right) = \sin \left( {5x - 3y} \right);\,\,\,\,\,\,P\left( {3,5} \right);\,\,\,\,\,\,\,{\bf{u}} = \frac{3}{5}{\bf{i}} - \frac{4}{5}{\bf{j}} \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( {\sin \left( {5x - 3y} \right)} \right) \cr & {f_x}\left( {x,y} \right) = \cos \left( {5x - 3y} \right)\left( 5 \right) \cr & {f_x}\left( {x,y} \right) = 5\cos \left( {5x - 3y} \right) \cr & and \cr & {f_y}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left( {\sin \left( {5x - 3y} \right)} \right) \cr & {f_y}\left( {x,y} \right) = \cos \left( {5x - 3y} \right)\left( { - 3} \right) \cr & {f_y}\left( {x,y} \right) = - 3\cos \left( {5x - 3y} \right) \cr & \cr & {\text{Calculate the gradient of }}f\left( {x,y} \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 & \nabla f\left( {x,y} \right) = 5\cos \left( {5x - 3y} \right){\bf{i}} - 3\cos \left( {5x - 3y} \right){\bf{j}} \cr & {\text{evaluate the gradient at the given point }}P\left( {3,5} \right) \cr & \nabla f\left( {x,y} \right) = 5\cos \left( {5\left( 3 \right) - 3\left( 5 \right)} \right){\bf{i}} - 3\cos \left( {5\left( 3 \right) - 3\left( 5 \right)} \right){\bf{j}} \cr & \nabla f\left( {x,y} \right) = 5\cos \left( 0 \right){\bf{i}} - 3\cos \left( 0 \right){\bf{j}} \cr & \nabla f\left( {3,5} \right) = 5{\bf{i}} - 3{\bf{j}} \cr & \cr & {\text{Calculate the directional derivative }}{D_{\bf{u}}}f\left( {x,y} \right){\text{ at }}P\left( {3,5} \right){\text{ in the direction of }}{\bf{u}} \cr & {D_{\bf{u}}}f\left( {x,y} \right) = \nabla f\left( {x,y} \right) \cdot {\bf{u}} \cr & {D_{\bf{u}}}f\left( {3,5} \right) = \nabla f\left( {3,1} \right) \cdot {\bf{u}} \cr & {D_{\bf{u}}}f\left( {3,5} \right) = \left( {5{\bf{i}} - 3{\bf{j}}} \right) \cdot \left( {\frac{3}{5}{\bf{i}} - \frac{4}{5}{\bf{j}}} \right) \cr & {\text{solving the dot product}} \cr & {D_{\bf{u}}}f\left( {3,1} \right) = \left( 5 \right)\left( {\frac{3}{5}} \right) + \left( { - 3} \right)\left( { - \frac{4}{5}} \right) \cr & {D_{\bf{u}}}f\left( {3,1} \right) = 3 + \frac{{12}}{5} \cr & {D_{\bf{u}}}f\left( {3,1} \right) = \frac{{27}}{5} \cr} $$