Calculate the directional derivative in the direction of v at the given point. Remember to normalize the direction vector or use Duf(P) = (1/||v||)∇fP · v.
f(x, y) = tan−1(xy), v = <1, 1>, P = (6, 7)
Let u = (v/||v||), v ≠ 0. Recall the formula for the directional derivative in the direction of v.
Duf(x, y) = ∇f(x, y) · u
= (1/||v||)∇f(x,y) · v
Calculate the unit vector, u, in the direction of v.
u = (v/||v||)
= <1, 1>/√
In order to calculate the gradient, we need to first calculate the partial derivatives. Find the partial derivatives, fx and fy at the point P = (6, 7) and round your answers to three decimal places. Since f(x, y) is a composition of functions, both of these derivatives will require the rule.
fx(x, y) =
A. B. C. D.
fx(6, 7) =
fy(x, y) =
A. B. C. D.
fy(6, 7) =
Calculate the gradient at point P, ∇fP.
∇fP = ∇f(6, 7)
= fx(6, 7), fy(6, 7)
= <, >
Find the directional derivative of f(x, y) in the direction of v at the point P. Round your answer to three decimal places.
Duf(6, 7) = ∇f(6, 7) · u
= (1/√2)<0.004, 0.003> · <1, 1>
= (1/√2)