In Connes' noncommutative geometry, the starting point is a spectral triple $(A,D,H)$ where $A$ is a commutative C* algebra, e.g. as in Connes "ON THE SPECTRAL CHARACTERIZATION OF MANIFOLDS" , where 5 conditions are stated which the triple must satisfy.

In order to construct coordinate charts on the $X = Spec(A)$, Connes proceeds with analysis of derivations $\delta_j$ on A, which satisfy exponentiability conditions (Lemma 3.3 in the above paper).

Question: where do these derivations come from?

They do not seem to be part of the package of the 5 conditions.

Is my understanding correct that Connes uses the spectrum of these derivations to identify local charts on X with $\mathbb{R}^p$?

For user's convenience, here are the 5 conditions:

(1) The n-th characteristic value of the resolvent of D is $O(n^{-1/p}$.

(2) $[[D,a],b]=0$ $\forall a,b \in A$

(3) For any $a\in A$, both $a$ and $[D,a]$ belong to the domain of $\delta^m$ for any integer m and $\delta(T) = [|D|,T]$

(4) There exists a Hochschild cycle $c\in Z_p(A,A)$ such that $\pi(c)=1$ for p odd, and $\pi(c)=\gamma$ is $\mathbb{Z}/2$ grading for p even

(5) The A-module $H_\infty = \cap Dom(D^m)$ is finite and projective. It also has hermitian structure $(|)$ defined as $<\xi,a\eta> = \int a(\xi|\eta)|D|^{-p}$