So it's said that differential forms provide a coordinate free approach to multivariable calculus. Well, in short I just don't get this, despite reading from many sources. I shall explain how it all looks to me.

Let's just stick to $\mathbb{R}^2$ for the sake of simplicity (maybe this is the down fall..). Picking some point $P=(x,y)\in\mathbb{R}^2$, we could ask about the directional derivative of some function $f:\mathbb{R}^2\rightarrow \mathbb{R}$, in direction $v=a\vec{i}+b\vec{j}$. This will be $$(\nabla \cdot v)|_P(f)=a\dfrac{\partial f}{\partial x}|_P+b\dfrac{\partial f}{\partial y}|_P =\underbrace{ (a\dfrac{\partial }{\partial x}|_P+b\dfrac{\partial }{\partial y}|_P)}_\text{$w_P$ }(f)$$ Where we can think of $w_P$ as an element of the tangent space at $P$. Now this in itself is a little weird; why have differential operators as a basis for something geometrical like a tangent space to a manifold? In any case, we apply these vectors to a function defined on our manifold, and we get the value we wanted out.

So who cares about differential forms? We just did all this without them. We could've done this by calculating $\mathrm{d}f$, in some basis $\mathrm{d}x, \mathrm{d}y$ (which is quite confusing), and then calculating $\mathrm{d}f(w_P)$, but what do we gain in doing it this way?

I mentioned I think the $\mathrm{d}x$'s are confusing. Well, $\mathrm{d}x$ is just the function $\mathrm{d}x(\frac{\partial}{\partial x})=1$ and 0 for any other differential operator - why write this as $\mathrm{d}x$, which had always previously been used to mean an infinitesimal increase in x?

Now I can understand caring about the dual of the tangent space. We are combining a vector in the tangent space with something and we're getting a scalar out of it - this something should then probably belong to the dual space. But if we're thinking of just the vector, then the function $f$ on the manifold needs to be encoded by the 1-form, right? Well, we can have 1-forms which aren't derivatives of any function on the manifold - what should it mean to combine such forms with tangent vectors?

And lastly, if we're writing all our forms in terms of $\mathrm{d}x$'s etc., where the $x$'s are exactly the coordinates of the manifold, then how exactly have we escaped dependence on coordinates? We're still essentially calculating with respect to a given coordinate system as in usual multivariable calculus!