Does derivative assigns diffrential?

Question

So there are 3 main definitions of derivation in 3 different contexts.

1. Calculus of one variable real functions. Say we have an everywhere differentiable function $f: \mathbb{R} \to \mathbb{R}$. Then its derivative is function $f': \mathbb{R} \to \mathbb{R}$. Its differential is linear map $df: \mathbb{R} \to \mathbb{R}$.
2. Functional analysis. Say we have a Banach spaces $E$ and $F$, and a function $f: E \to F$. Then for each point $x \in X$ we can define a Frechet derivative $f'(x)$ which is a continuous linear map $E \to F$. So we have an assignment $x \mapsto B(E, F)$.
3. Manifolds. Say we have manifolds $M$ and $N$ and a morphism $f: M \to N$. Then for each point $x \in M$ we can assign a differential $df_x$ which is a linear map of tangent spaces $df_x: T_x M \to T_{f(x)} N$.

Is it true that all of these definitions coincide in the context of calculus?

To be more precise:

So is it true that a derivative is in fact an assignment of differential to each point? Where differential is morphism of tangent spaces?

I use term assignment to emphasize that tangent spaces may be different for each point. It would be beautiful if there is a functor of some sort, but I can't figure it out.

Here are my thoughts:

• There is a distinct notion of derivative and differential in the context of calculus, but no differential of function of Banach spaces, nor derivative of morphism of manifolds.
• Frechet derivative can be identified with usual derivative because $B(\mathbb{R}, F) \cong F$ (isometric isomorphism).
• Differential of morphisms coincide with usual differential because $T_x \mathbb{R} \cong \mathbb{R}$.
• So $T_x E \cong E$ for Banach spaces? (Some kind of synthetic-algebraic definition needed here.) Is there a notion of differential forms on Banach spaces, e.g. when doing integrals?
• Is there some geometric structure on tangent spaces? I mean from algebraic view they all have same dimension $T_x M \cong T_y M \cong \mathbb{R}^m$, so we may define a function $f': M \to L(\mathbb{R}^m, \mathbb{R}^n)$? (Here $\dim M = m, \dim N = n$.)

Bonus: what about TVS?