Directional derivative of a vector field in the direction of another vector field

Question

So, I was reading some introductory things about differential geometry, and just discovered the big gap I have in analysis. For instance, I do understand what is means to ask for the directional derivative of a smooth real valued function $f: \mathbb{R}^n \rightarrow \mathbb{R}$, in $u \in \mathbb{R}^n$ direction, and how does this generalizes to the case of vector fields of the form $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$, where we have $\nabla_u F(p) = \lim_{t \to 0} \frac{F(p+tu)-F(p)}{t},$ as the directional derivative. Now, I'm looking for the directional derivative of a vector field say $X$, in the direction of another vector field $Y$, but I can't find any reference or online handout with this sort of things. I presume that the definition should be look like $$\nabla_Y X(p) = \lim_{t \to 0} \frac{X(p+tY)-X(p)}{t},$$ but I need to see it written anyway, so any comments, references or online notes concerning those things are welcome.