Answer
$\sqrt 3$
Work Step by Step
Since, $f(x,y,z)=xyz$
We know that $D_u f = \nabla f \cdot u$
So, $\nabla f =yz i +xz j +xy k$
and $\nabla f(1,1,1) =(1)(1) i +(1)(1) j +(1)(1) k=i+j+k$
So, $D_u f (1,1,1)=| \nabla f|_{max}=\sqrt {1^2+1^2+1^2}=\sqrt 3$
