Answer:
Directional derivative = 1/√2
Step-by-step explanation:
We are given f(x, y) = y cos(xy)
Now, we know that;
∇f(x, y) = ycos xy
Thus, applying that to the question, we have;
∇f(x, y) = [-y² sin xy, cos (xy) - xy sin xy]
At coordinates (0,1),we now have;
∇f(0, 1) = [-1²•sin0, (cos 0) - 0]
∇f(0, 1) = [0, 1]
Now, unit vector indicated by the angle θ is given as; u = [cos θ, sin θ]
From the question, since θ = π/4, thus
u = [cos π/4, sin π/4]
Cos π/4 in surd form is; 1/√2
Also, sin π/4 in surd form is; 1/√2
So, u = [1/√2, 1/√2]
Directional derivative = [∇f(0, 1)] • u
= [0, 1] × [1/√2, 1/√2] = 0 + 1/√2 = 1/√2
