Hadamard's lemma, in one dimension, says for any smooth function $f \colon \mathbf R \rightarrow \mathbf R$ there is a first-order expansion of $f$ at $0$: $f(x) = f(0) + xg(x)$ where $g \colon \mathbf R \rightarrow \mathbf R$ is smooth. Explicitly, $g(x) = \int_0^1 f'(tx)\,dt$. (The analogue in higher dimensions is on the Wikipedia page for Hadamard's lemma.) My question is: where/when did Hadamard first present this result, or is it not due to him at all (Stigler's law)?
I looked through the four volumes of Hadamard's Oeuvres and the only thing I could find resembling this result was in his 1892 PhD thesis on analytic functions. Sections 32-35 contain some integrals of the form $\int_0^1 V(t)f(tx)\,dt$ for different choices of $V(t)$. Perhaps these motivated someone else (Whitney?) later on when the general foundations of manifolds were laid down in the 20th century.