Example 1

Tag 0669, permanent. Cite it as [Bourbaki, Tag 0669].

Let A be an $\mathbf{R}$ -algebra of differentiable mappings of $\mathbf{R}$ into $\mathbf{R}$ and let $x_0$ be a point of $\mathbf{R}$ ; $\mathbf{R}$ can be considered as an A-module with the external law $(f, a) \mapsto f(x_0)a$ . Then the mapping $f \mapsto Df(x_0)$ is a derivation, since (Functions of a Real Variable, I, § 1, no. 3)

(D(fg))(x_0) = (Df(x_0))g(x_0) + f(x_0)(Dg(x_0)).*

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Corollary, § 10

Cites

Functions of a Real Variable, I, § 1, no. 3 not yet on this site: FVR I