There's the following observation with a very short proof:
Observe that if $\varphi : M \to M$ is a diffeomorphism, $v \in T_p M$ and $f$ is a differentiable function in a neighborhood of $\varphi(p)$ , we have
$$
(d\varphi(p) f) \varphi (p) = v(f \circ \varphi)(p)
$$
Indeed, let $\alpha : (-\epsilon, \epsilon) \to M$ be a differentiable curve with $\alpha'(0) = v$ and $\alpha(0) = p$. Then
$$
(d \varphi (v) f) \varphi (p) = \left. \frac{d}{dt} (f \circ \varphi \circ \alpha) \right|_{t=0} = v(f \circ \varphi)(p)
$$
I don't get the very last line, I suppose it follows from the definition given of tangent vector, but I really can't work out the details.
Could you explain more in detail that line?
Thank you (P.S. here $M$ is a differentiable manifold).
Update : In the statement of this theorem $\beta$ is defined as $\beta = \varphi \circ \alpha$. Given a curve $\alpha : (-\epsilon,\epsilon) \to M$ (differentiable manifold) a tangent vector is defined as
$$
\alpha'(0) f = \frac{d}{dt} (f \circ \alpha).
$$
Using $\beta$ instead of $\alpha$ we have
$$
\beta'(0)f = \frac{d}{dt}(f \circ \beta) = \frac{d}{dt}(f \circ \varphi \circ \alpha)
$$
but $\beta'(0) = d \varphi_p(v)$ therefore $\beta'(0) f = d \varphi_p(v) f$ and therefore we have
$$
d \varphi_p(v) f = \frac{d}{dt}(f \circ \varphi \circ \alpha)
$$
Therefore what I think I'm actually missing is the meaning of the notation $ \left( d \varphi_p(v) f \right) \varphi(p)$
More details
Just to clarify, my previous update is my interpretation of the notation used, I've noticed there's some inconsistencies (I think) in the notation used throughout the book (at least for chapter 0).
1) In definition 2.6.
The author defines a tangent vector to a differentiable manifold $M$ as an operator $\alpha'(0) : \cal{D} \to \mathbb{R}$, where $\cal{D}$ is the set of differentiable functions defined on $M$. Such description makes very clear to me the meaning of the notation $\alpha'(0)f$, If I view it as an operator. Moreover we have the expression given
$$
\alpha'(0) f = \frac{d}{dt}(f \circ \alpha)
$$
The right handside of the expression in terms of computation is clear to me, because of the parameterization.
2) In proposition 2.7, given a differentiable map $\varphi$ between two differentiable manifolds $M_1^n, M_2^m$ and the differential $d \varphi_p(v)$ is defined. To define this differential we need a $p \in M_1^n$ and a tangent vector $v \in T_p M_1^n$. The differential is a map from $T_p M_1^n$ to $T_{\varphi(p)} M_2^m$, and therefore it acts as an operator from the set of differentiable functions on $M_2^m$ to $\mathbb{R}$.
Thefore the meaning of the notation $ d \varphi_p(v) f $ is still clear to me.
3) My original question concern page 26 of the book, where the notation $(d \varphi(v) f)\varphi(p)$ is introduced. The reason I get confused is in the first place I think there's a mistake in the notation used because if you compare $d \varphi_p(v) f$ against $(d \varphi (v) f) \varphi(p)$ there's some difference. I don't know for example what $d \varphi(v)$ means, but I do know what $d \varphi_p(v)$ means. Also $d \varphi_p(v) f$ returns a real value by definition and this doesn't not seem evident to me from $(d \varphi(v) f) \varphi(p)$.
I hope I clarified what my issue is.
Best Answer
The meaning of the notation (though it is clumsily written) is that of taking the derivative of $f$ at the value $\varphi(p)$ with respect to the vector $d\varphi_p(v)$ since in this case $v\in T_pM$ and therefore $d\varphi_p(v)\in T_{\varphi(p)}M$. A much clearer way to convey the same idea would be to write $\left. \left (d\varphi_p(v)f\right ) \right |_{\varphi(p)}$.
I've worked through a lot of do Carmo, and I can comfortably say as a fair warning that his notations are less clear than other treatments of manifold theory and Riemannian geometry. In my experience, do Carmo's chapter 0 is a poor place to learn the subject for the first time. I'd highly recommend comparing his text to analogous statements in other books. For example, consider using John Lee's Introduction to Smooth Manifolds as a companion text. It might be easier to come back to do Carmo and learn Riemannian geometry there after learning about manifolds elsewhere.