The definition of weak derivative in the book Partial Differential Equations by Evans is stated as follows:
Suppose $u,v \in L_{loc}^1(U)$, and $\alpha$ is a multiindex. We say that $v$ is the $\alpha$th-weak partial derivative of $u$, written \begin{equation*} D^\alpha u=v, \end{equation*} provided \begin{equation*} \int_U u D^\alpha \phi d x=(-1)^{|\alpha|} \int_U v \phi d x \end{equation*} for all test functions $\phi \in C_c^\infty(U)$.
I have tried to prove that $u$ has the $\beta$th-weak partial derivative for any $\beta \leq \alpha$ if it has the $\alpha$th-weak partial derivative but failed. The reason why I tried to do this is that the text followed in this book seems to assume it holds, though not that obvious for me at the first glance.
PS: Suppose $\alpha=(\alpha_1,\dots,\alpha_n)$, $\beta=(\beta_1,\dots,\beta_n)$ are multiindices. By $\beta \leq \alpha$ we mean $\beta_i \leq \alpha_i$, $i=1,\dots,n$.
