I attempt to understand the computation of the differential of a smooth map between two smooth manifolds using smooth curves from Loring Tu's An Introduction to Manifolds (Second Edition, page no. 95). The corresponding section of the book is given below. 
I understand the proof and can derive up to the following expression: $$F_{*, p} \, (X_p) = (F \circ c)_{*,0} \left( \left.\frac{d}{dt}\right\vert_0 \right).$$
But I don't understand how we get the last line: $$\cdots = \left.\frac{d}{dt}\right\vert_0 (F \circ c)(t).$$
I proceeded as follows. As $F_{*, p} \, (X_p): C_{F(p)}^{\infty}(M) \to \mathbb{R}$ is a derivation at the point $F(p) \in M$, I choose an element $[f]_{F(p)}$ of of the algebra $C_{F(p)}^{\infty}(M)$ and compute the followings.
$$\left(F_{*, p} \, (X_p)\right)(f) = \left( (F \circ c)_{*,0} \left( \left.\frac{d}{dt}\right\vert_0 \right)\right)(f) = \left.\frac{d}{dt}\right\vert_0 (f \circ F \circ c)(t),$$ where in the last step I have used the definition of a differential of a smooth function between two smooth manifolds.
Now I am not sure how to proceed any further to get the desired expression in the last line of the book.