The directional derivative of function f(x,y,z)at point (x0,y0,z0) in the direction of v is ⟨∇f(x0,y0,z0),v⟩, where ⟨,⟩ is the scalar product in R3.
The gradient of f(x,y,z) is ∇f(x,y,z)=(6x,−3y2z2,−2zy3) and is equal to (6,−12,16) at given point x0=(1,2,−1).
Hence, the directional derivative at x0 in the direction of (1,1,1) is ⟨(6,−12,16),(1,1,1)⟩=10.
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