Calculus 3 Rate of Change

Question

Find the direction in which the maximum rate of change occurs for the function f(x,y)=3xsin(xy) at the point (3,1). Give your answer as a unit vector.

Answer 1

∇f(x,y) = f_xi +f_yj where f_x = 3sin(xy) +3xycos(xy) f_y= 3x²cos(xy).Then ∇f =3(sinxy +xycosxy)i +3x²cosxyj . At the point (3,1) ∇f = 3(sin3 + 3cos3)i + 27cos3j so ∇f = 3(.1409 -2.97)i - 26.7j = -8.876i -26.7j. |∇f| =(78.795 +712.89)^1/2 = 28.137. The unit vector is (-8.876i - 26.7j)/ 28.137.