So in the context of differential geometry and covariant derivatives, I've seen the following formula for computing directional derivatives of a vector field $\mathbf X$.
$$D_{\mathbf V} \mathbf X = (\mathbf X(\mathbf{\alpha}(t)))'(0)$$
where $\mathbf \alpha$ is a curve with $\alpha(0) = P$ and $\alpha ' (0) = \mathbf V$.
But how does one use this in practice? Using the chain rule, you'd want the derivative of $\alpha$ times the derivative of $\mathbf X$ of $\alpha$. But if you use the dot product, that would give you a scalar result. It seems like it should be a vector result, since the derivative of a vector function has the same amount of dimensions as the original.
So how do you use this formula in practice?