Differential on a manifold I got confused with the differential of a map on a manifold. I try to wrap up what I learned and what I am confused about. Let $X$ be a manifold and $c_1,c_2: (-\epsilon, \epsilon) \to X$ two curves satisfying $c_1(0) = x = c_2(0)$ for an $x \in X$. A tangent vector at $x$ is an equivalence class of curves satisfying the relation $c_1 \sim c_2 \Longleftrightarrow D(\varphi \circ c_1)(0) = D(\tilde{\varphi} \circ c_2)(0)$ where $(U, \varphi)$ and $(V,\tilde{\varphi})$ are charts around x and the tangent space of $X$ at $x$ is defined as the set of all these equivalence classes. If I have a map $f: X \to Y$ where $Y$ is another manifold, I can define the tangent map $T_xf: T_xX \to T_yY$ where $[c] \mapsto [f \circ c] \in T_yY$. But how can I define the derivative of $f$ at $x$ in this setup? I don't want to to this with derivations. 
Since the local representation of $f$ is $(\psi \circ f \circ \varphi^{-1}). (\varphi(x))$ cant I just differentiate this and obtain $Df$ in this way i.e. $Df(x) = D(\psi \circ f \circ \varphi^{-1})(\varphi(x))$? And is this related to the tangent map? If yes, what happened to $[f \circ c]$? I am sorry for my (maybe confusing) explanation but I can't get my head around this. 
 A: Actually you can define the differential in 3 different way:


*

*Through derivations, identifying (as you've done) functions that
locally are alike since the differential is local notion and so you
won't have anything to distinguish to functions which behave in the
same manner locally but then change outside an open set;

*Bringing everything back to ordinary differential of functions
between $R^m$ and $R^n$ and then simply treat it with ordinary
calculus instruments;

*Thinking tangent vectors as velocities of curves and then
calculating the velocity of all the possible curves in a point of
the manifold.


Now, these 3 picture coincide on $R^n$ coincide so you might want to try them on something concrete as a half sphere or a paraboloid, but the most general picture (which can be extended in an easy way to all manifolds) happens to be first one with derivations. Once you get used you will understand that is indeed the most useful. 
You just have to convice yourself that in $R^n$ there's an identification between directional derivatives, tangent vectors and possible velocities of a curve in a point. Once you got it you just have to convince yourself that what you're really looking for are directional derivatives and that a generalization for them are Derivations.
Sorry to be so vague, if you need some specific clarification tell me in the comments and I will edit the answer.  
