Find the directional derivative of f(x,y,z) = -5x^2 -2y^2 -4z^2 at the point (-1, -2, 1) in the direction of the origin.

Question

I got the partial derivatives:

fx=-10x

fy=-4y

fz=-8z

then I got that the gradient of f is <10, 8, -8>

I am struggling to find the direction of the vector that you have to do the dot product with to get the directional vector.

Answer 1

A vector in the required direction is (0, 0, 0) − (-1, -2, 1) or (1, 2, -1).

The unit vector in that direction is u equal to [1/√(1² + 2² + (-1)²] times (1, 2, -1) or [1/√6](1, 2, -1).

Take the gradient of f(x, y, z) as ∇f equal to {∂f/∂x, ∂f/∂y, ∂f/∂z}, here equal to {-10x, -4y, -8z}.

Now evaluate ∇f at (x, y, z) equal to (-1, -2, 1) to gain {-10(-1), -4(-2), -8(1)} or {10, 8, -8}.

From the above analysis, the directional derivative sought is ∇f • u or {10, 8, -8} • (1/√6, 2/√6, -1/√6)

or [10/√6 + 16/√6 + 8/√6] or 34/√6.