I am totally new to the subject differential geometry, and that probably reflects itself in the naive question that I'm trying to formulate. I hope this question does not get closed because of this.

Let $M$ be a smooth manifold, equipped with an affine connection $\nabla$. Given two vector fields $X$ and $Y$, there are three ways we can "differentiate" $X$. The first one is to take Lie derivative $\mathscr{L}_Y X$, the second one is to do $\nabla_Y X$ and the third one, the most naive one, is just to take $DX: TM \to TTM$, which is just the total differential of $X$, viewed as a map $X: M\to TM$.

Note that the most naive way is very similar to taking exterior derivative of a function, and so, philosophically, it should be thought of as the "total derivative" and all others are somehow just partial derivatives.

This makes me believe (naively) that there is conceptually clean way to work from the most naive way of differentiating and deduce everything else (exterior derivatives (of higher wedges as well), Lie derivatives and connections) from it. I hope someone can clarify this for me.

Thank you.