The derivative of [tex]f(x,y,z)[/tex] in the direction of a vector [tex]\mathbf u[/tex] is
[tex]\nabla_{\mathbf u}f=\nabla f\cdot\dfrac{\mathbf u}{\|\mathbf u\|}[/tex]
With [tex]f(x,y,z)=xyz[/tex], we get
[tex]\nabla f=(yz,xz,xy)[/tex]
and [tex]\mathbf u=(-1,-2,2)[/tex],
[tex]\|\mathbf u\|=\sqrt{(-1)^2+(-2)^2+2^2}=3[/tex]
Then
[tex]\nabla_{(-1,-2,2)}f(3,3,9)=(27,27,9)\cdot\dfrac{(-1,-2,2)}3=-21[/tex]
