F(x,y,x)=tan(x+2y+3z), (-5,1,1)

Find the maximum rate of change of f at the given point and the direction in which it occurs. f(x,y,x)=tan(x+2y+3z), (-5,1,1)

This will occur in the direction of ∇f(-5, 1, 1).

∇f =
==> ∇f(-5, 1, 1) = <1, 2, 3>.

Normalizing yields the unit vector
u = ∇f(-5, 1, 1) / ||∇f(-5, 1, 1)|| = <1, 2, 3>/√14.

So, the maximum rate of change equals
∇f(-5, 1, 1) · u = √14.

I hope this helps!