Question: In differential geometry, there are several notions of differentiation, namely:

* Exterior Derivative, $d$
* Covariant Derivative/Connection, $\nabla$
* Lie Derivative, $\mathcal{L}$.

I have listed them in order of appearance in my education/in descending order of my understanding of them. Note, there may be others that I am yet to encounter. 

Conceptually, I am not sure how these three notions fit together. Looking at their definitions, I can see that there is even some overlap between the collection of objects they can each act on. I am trying to get my head around why there are (at least) three different notions of differentiation. I suppose my confusion can be summarised by the following question. 

> 
> What does each one do that the other two can't?
> 
> 
> 

I don't just mean which objects can they act on that the other two can't, I would like a deeper explanation (if it exists, which I believe it does). In terms of their geometric intuition/interpretation, does it make sense that we need these different notions?

---

Note, I have put the reference request tag on this question because I would be interested to find some resources which have a discussion of these notions concurrently, as opposed to being presented as individual concepts.

Answer: 
A very short answer:

In finite dimensions and at least in characteristic 0, the equation $$\operatorname{d} \omega(x, y) = y(\omega(x)) - x(\omega(y)) - \omega([x, y])$$ allows you to define $[-,-]: V \wedge V \to V$ starting from $\operatorname{d}: V^\* \to V^\* \wedge V^\*$, and vice versa.

Furthermore, you can prove that conditions $[[x, y], z] + [[y, z], x] + [[z, x], y] = 0$ and $d^2 = 0$ are equivalent under this correspondence.

Now, unfortunately this doesn't apply to infinite-dimensional case that is general Lie derivative and exterior derivative, and I don't know how relevant this is to general Lie derivative and exterior derivative, but it's very nice to know this relation at least in purely algebraic case.

<https://en.wikipedia.org/wiki/Lie_coalgebra>
Here is my humble take on this very interesting question. I think of the exterior differentiation $d$ as some kind of infinitesimal Stoke's theorem, so to speak. Namely, if $\alpha$ is a $k$-form, then $d\alpha$ is of course a $k+1$-form, and it can be defined at some point $p$, in some directions determining a $k+1$ dimensional subspace of the tangent space $T\_p$, by essentially taking a small $k+1$ dimensional closed ball, and associating to it the integral of $\alpha$ on the boundary $k$-dimensional sphere, and then normalizing this integral properly and then letting the radius of the ball go to $0$. This is similar to what one sees in some Mathematics for Engineering books actually, albeit in some special cases (divergence, curl etc.).

I think of a covariant derivative mathematically using parallel transport and holonomy, and physically, for a connection on some unitary vector bundle, as some kind of potential (similar to the electromagnetic potential, but of course, it could be more complicated).

I think of the Lie derivative along $X$ as some kind of derivative along the (local in "time") flow induced by $X$. I think of $[X,Y]$ at some point $p$ as the vector going from $p$ to the point obtained by flowing along $X$ for some time $t$, then $Y$ for some time $s$, then $-X$ for some time $t$, then $-Y$ for some time $s$. For this to be made rigorous, one has to divide by $st$, and then take the limit as $s$ and $t$ go to $0$. I learned about this intuitive picture while reading Sir Roger Penrose's "Road to Reality" (which I enjoyed reading).

Philosophically speaking, I like to think of these derivatives using the most intuitively appealing (for me) definitions, which are usually close to the historical ones. However, when it comes to computing them, of course, this is almost never the easiest approach: then one can resort to formulas such as Cartan's magic formula $L\_X = d \circ \iota\_X + \iota\_X \circ d$, and so on.