This is probably beyond my high-school level competence, but I'm looking at page 112 of Spivak's A Comprehensive Introduction To Differential Geometry and wondering how he gets from$$\frac{d\left(f\left(c\left(t\right)\right)\right)}{dt}$$to $$df\left(\frac{dc}{dt}\right).$$
Highlighted below. He's comparing the classical and modern formulations of $df$.
I'm not even sure what the second term means. Is it saying $df$ is a function of $dc/dt$ or is it saying $df$ is multiplied by $dc/dt$? I've fiddled around with the chain rule, but haven't made any progress. Thanks.
This is all clearly explained in the book, but I’ll spell it out again. For Spivak $df$ is a section of the cotangent bundle. That means it is a function from $M$ to $T^*M$, so it assigns to every point $p$ a one-form $df(p)$ that lives in the cotangent space $T^*_pM$ to $M$ at $p$. This one-form $df(p)$ is thus, itself, a linear function from $T_pM$ to $\mathbb{R}$; it operates on a tangent vector to $M$ at $p$ and gives a scalar.
In particular, this one-form $df(p)$ operates on the tangent vector $X\in T_pM$ to give the scalar $X(f)$. Why is $X(f)$ a scalar? This is where you have to understand Chapter 3 on tangent vectors; it is basically the directional derivative of the scalar-valued function $f$ at $p$ taken in the direction of the tangent vector $X$.
To be formal, you can think of $X$ as the tangent vector $c’(t_0)$ to a parametrized curve $c:\mathbb{R}\to M$ with $c(t_0)=p$. But what does “$c’(t_0)$” really mean? It means the pushforward $c_*(d/dt|_{t_0})$ by the map $c$ of the standard tangent vector basis on $\mathbb{R}$. The final step is the important relation at the top of page 80, which tells us how the pushed-forward vector acts; it says (using different letters)
$$(c_*(d/dt|_{t_0}))(f)=d/dt(f\circ c)|_{t_0}$$
Thus, we’ve shown
$$[df(p)](c’(t_0))=d/dt(f\circ c)|_{t_0}$$
If you declutter the notation by dropping the explicit dependence on $t_0$ and thus $p$ as well (as Spivak explicitly says he is doing in the first line of page 110), then this says that the two expressions you’ve highlighted in yellow are equal.