Distinguish from Hadamard's formula in Lie theory, which is also often called Hadamard's lemma.
(Hadamard lemma)
For every smooth function $f \in C^\infty(\mathbb{R})$ on the real line there is a smooth function $g$ such that
This function $g$ is also called a Hadamard quotient.
It follows that $g(0) = f'(0)$ is the derivative of $f$ at 0. By applying this repeatedly the lemma says that $f$ has a partial Taylor series expansion whose remainder $h$ is a smooth function:
More generally, for smooth functions on any Cartesian space $\mathbb{R}^n$ the lemma says that there are for each $f \in C^\infty(X)$ $n$ smooth functions $g_i$ such that
So at the origin these smooth functions compute the partial derivatives of $f$
Holding $x$ fixed, put $h(t) = f(t x)$. Then
where the second equality uses the chain rule. The lemma follows by putting
The Hadamard lemma implies that the derivations of the algebra $C^\infty(X)$ of smooth functions on a smooth manifold are in bijection with the vector fields on $X$. See derivations of smooth functions are vector fields for details.
A more abstract way to state the Hadamard lemma (and a bit more) is to say that smooth function rings form a Fermat theory. As such the Hadamard lemma is a crucial ingredient for well-adapted models of synthetic differential geometry.
The Hadamard lemma is due to Jacques Hadamard.
The Hadamard lemma is what makes the standard convenient models for synthetic differential geometry tick. Its role in this respect can be seen from proposition 1.2 on in