Question

Find the directional derivative of the function at point P in the direction of Q. f(x, y, z) = y/(x + z) , P(2, 1, −1), Q(−1, 7, 0)

Answer 1

To find the directional derivative of the function at point P in the direction of Q, calculate the dot product of the gradient vector of the function and the unit vector in the direction of Q.

Step-by-step explanation:

To find the directional derivative of the function at point P in the direction of Q, we need to calculate the dot product of the gradient vector of the function and the unit vector in the direction of Q.

The gradient vector of the function f(x, y, z) = y/(x + z) is ∇f = (y/(x + z)^2, 1/(x + z), -y/(x + z)^2).

The unit vector in the direction of Q is Q/|Q| = (-1/√51, 7/√51, 0/√51).

Taking the dot product of ∇f and Q/|Q| gives us the directional derivative:

DQf = ∇f · Q/|Q| = (-y/(x + z)^2)(-1/√51) + (1/(x + z))(7/√51) + (-y/(x + z)^2)(0/√51) = y/(√51(x + z)^2) + 7/(√51(x + z)).