I'd like you to check my answer. This is the most common exercise about directional derivatives...

Let $\displaystyle f(x,y,z)=xyz$, $\displaystyle \bold u =( \cos (\alpha) \sin (\beta) , \sin (\alpha) \sin (\beta) , \cos (\beta) )$ and $\displaystyle \bold x = (1,0,1)$.

Find the directional derivative of $\displaystyle f$ in the direction of $\displaystyle \bold u$, in the point $\displaystyle \bold x$.

Here's my work :

$\displaystyle \frac{\partial f}{\partial \bold u} (\bold x)= \nabla f( \bold x) \cdot \bold u = \sin (\alpha) \sin (\beta)$.

I think it is right, but I'm unsure.