Answer to Question #223880 in Chemical Engineering for Lokika

Question

Find the directional derivative of f(x,y,z)=xyz^{2}+xz along the tangentto the curve x=t,y=t^{2},z=t^{3} at the point (1,1,1)

Answer 1

We begin with the graph of a surface well-defined by the equation z=f(x,y). Given a point (a,b) in the domain of f, we select a path to travel from that point. We measure the direction by an angle θ, which is measured anticlockwise in the x, y-plane, beginning at zero from the positive x-axis. The distance we travel is h, and the direction we travel is given by the unit vector u=(cosθ)i+(sinθ)j. Consequently, the z-coordinate of the second point on the graph is given by z=f(a+hcosθ,b+hsinθ).