I am following this introductory video on differential forms by Prof. Michael Penn. At 5:44, he defines the following linear map (which I guess is a differential form:
$ \langle dx,dy\rangle(v) : T_p M \to \mathbb{R}^2 , \langle dx(v),dy(v)\rangle = \langle dx(v),dy(v)\rangle$
After his example at 8:17 involving calculation of tangent vector component of a parabola, I could easily grasp the above definition as the idea that the map takes in the geometric tangent vector and spits out it's components with respect to a basis (I am guessing cartesian basis but he didn't mention explicitly).
Here comes the part I am confused, starting from 9:20 he equates the component functions of the above mentioned linear map evaluated a point to the tangent vector of parabola given by the parameterization $y=x^2$.
Symbolically, At a point $(a,a^2)$ the tangent vector is equated as:
$$\langle 1,2a\rangle = \langle dx(v) , dy(v)\rangle$$
And then he equates the components:
$$ 1=dx(v)$$
And,
$$ dy(v) = 2a(1) = 2adx(v)$$
Or,
$$ \frac{dy(v)}{dx(v)} = 2a$$
And states that the LHS is just the definition of derivative. In the beginning the prof notes that the expression is supposed to denote a linear map, now at the end it when both the map components evaluated are taken as a ratio, it turns out into in the derivative. Why is the differential form both a linear map and a differential?