Calculate the directional derivatives of $\displaystyle f(x,y,z)=x^2y+y^2z+z^2x$ at (1,2,-1) in the direction $\displaystyle u=<2/3,-2/3,1/3>.$
First find the Gradient:
$\displaystyle \vec{\nabla} f\!\left(x,y,z\right)=\frac{\partial}{\partial x}\left(x^2y+y^2z+z^2x\right)\mathbf{i}+\frac{\par tial}{\partial y}\left(x^2y+y^2z+z^2x\right)\mathbf{j}$ $\displaystyle +\frac{\partial}{\partial z}\left(x^2y+y^2z+z^2x\right)\mathbf{k}=\left(2xy+ z^2\right)\mathbf{i}+\left(x^2+2yz\right)\mathbf{j }+\left(y^2+2zx\right)\mathbf{k}$
Thus, the gradient at $\displaystyle \left(1,2,-1\right)$ would be $\displaystyle \vec{\nabla} f \!\left(1,2,-1\right)=5\mathbf{i}-3\mathbf{j}+2\mathbf{k}$
Now, the directional derivative is equal to $\displaystyle \vec{\nabla} f\!\left(1,2,-1\right)\cdot\mathbf{u}=\left<5,-3,2\right>\cdot\left<\tfrac{2}{3},-\tfrac{2}{3},\tfrac{1}{3}\right>=\tfrac{10}{3}+2+\ tfrac{2}{3}=\boxed{6}$
Does this make sense?