Final answer:
To compute the directional derivatives of a function, we use the gradient vector and partial derivatives. By substituting the values into the formula, we can calculate the directional derivatives for the given functions. For example, we can find the directional derivative of f(x, y, z) = x^2 + 2y^2 - 3z^2 at a given point in a specific direction.
Step-by-step explanation:
Solution:
To find the directional derivatives of a function at a given point in a specific direction, we need to use the gradient vector.
The directional derivative of a function f(x, y, z) at a point (x0, y0, z0) in the direction of a unit vector = ai + bj + ck is given by:
Df(x0, y0, z0) = a ∂f/∂x + b ∂f/∂y + c ∂f/∂z
where ∂f/∂x, ∂f/∂y, and ∂f/∂z are the partial derivatives of f(x, y, z) with respect to x, y, and z, respectively.
By substituting the values of (x0, y0, z0) and (, , ) into the formula, we can compute the directional derivatives for the given functions.
For example, if we have a function f(x, y, z) = x2 + 2y2 - 3z2 and we want to find the directional derivative at the point (1, 2, 3) in the direction of the unit vector = 3i - j + 2k, we substitute x0 = 1, y0 = 2, z0 = 3, a = 3, b = -1, and c = 2 into the formula to compute the directional derivative Df(1, 2, 3).