Lemma 6

Let $k$ be a natural number. Let $f\in \mathscr{C}^k(U)$ and let $\lambda$ be the distribution associated with $f$. For every $\beta \in \mathbf{N}^n$ such that $|\beta |\leqslant k$, the distribution $\partial^{\beta}\lambda$ is the distribution associated with the function $\partial^{\beta}f$.

By induction on $k$, it is enough to prove this property when $\beta$ satisfies $|\beta |= 1$, and one may even suppose that $\beta = (0, . . . ,0,1)$. Since distributions define a sheaf (prop. 5 of IV, p. 204), it is enough to verify the assertion when there exist an open set $V\subset \mathbf{R}^{n-1}$ and an open interval $I\subset \mathbf{R}$ such that $U = V\times I$.

For every test function $\varphi \in \mathscr{D}$(U), one has by definition

from the Lebesgue–Fubini theorem (INT, V, p. 96, § 8, n$^o4$, th. 1). By integration by parts (FVR, II, p. 10), one has

since $t\mapsto \varphi (y, t)$ has compact support in I. One therefore obtains