For a scalar function f(x, y, z) = x² + 3y² + 2z², the directional derivative at the point p(1, 2, – 1) in the direction of a vector ^$\acute{i}$– ^$\acute{j}$+^{$2 \bar{k}$} is

The directional derivative is the dot product of the gradient vector and the unit vector in the specified direction. Gradient at (1, 2, -1) is (2, 12, -4). The direction vector is (1, -1, 2) with magnitude sqrt(1^2 + (-1)^2 + 2^2) = sqrt(6). Dot product is (2*1 + 12*-1 + -4*2) / sqrt(6) = (2 - 12 - 8) / sqrt(6) = -18 / sqrt(6) = -3 * sqrt(6).