Directional derivative of x²yz+4xz²

The directional derivative of φ along a vector d at a point is ∇φ · d̂, with d̂ = d/|d|.

Gradient

φ = x²yz + 4xz²

∂φ/∂x = 2xyz + 4z²
∂φ/∂y = x²z
∂φ/∂z = x²y + 8xz

At (1, −2, −1):

∂φ/∂x = 2(1)(−2)(−1) + 4(1) = 4 + 4 = 8
∂φ/∂y = (1)(−1) = −1
∂φ/∂z = (1)(−2) + 8(1)(−1) = −2 − 8 = −10
∇φ = 8i − j − 10k

Unit vector

d = 2i − j − 2k, |d| = √(4 + 1 + 4) = 3, d̂ = (2i − j − 2k)/3.

Directional derivative

∇φ · d̂ = (8·2 + (−1)(−1) + (−10)(−2))/3 = (16 + 1 + 20)/3 = 37/3.

Answer: 37/3 ≈ 12.33 (φ increases in that direction).