Say we have a smooth function
$f(x,y,z)=z^2+7x^2e^{yz}$
and a vector field
$X(x,y,z)=(-x,y,z)$
Now say we want to calculate the directional derivative $X(p)[f]$ at the point $p=(0,1,2)$. Then is this the correct method :
$X(p)[f]=\tfrac{d}{dt}|_{t=0}f(\alpha(t))$
$\alpha(t)=p+tX=(0,1,2)+t(-x,y,z)=(0,1,2)+t(0,1,2)=(0,1+t,2+2t)$
$\therefore f(\alpha(t))=(2+2t)^2=4+8t+4t^2$
$\tfrac{df(\alpha(t)}{dt}=8+8t$
so
$\tfrac{df(\alpha(t)}{dt}|_{t=0}=8$
which is just a scalar number.
Note: the reason I'm asking is because i'm studying for an exam an I wasn't sure whether $p+tx=(0,1,2)+t(-x,y,z)=(-tx,1+ty,2+2z)$ or whether it was what I stated above , above produces a scalar this method produces a smooth function.