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If $u$ is a locally integrable function on some open set $U \subset \mathbb{R}^n$, and $\alpha$ is a multiindex, we say that $v$ is the $\alpha^{\text{th}}$-weak partial derivative of $u$, and write $$D^{\alpha} u = v$$ if for all test functions $\varphi \in C_c^{\infty}(U)$, it is true that

$$\int_U uD^{\alpha} \varphi dx = (-1)^{|\alpha|} \int_U v \varphi dx$$

In the particular case that $u \in L^1([a, b])$, then $v$ is the weak derivative of $u$ if

$$\int_a^b u \varphi' dx = - \int_a^b v \varphi dx$$

for all infinitely differentiable $\varphi$ such that $\varphi(a) = 0 = \varphi(b)$.

This can be viewed as a generalization of the usual integration by parts formula, and can be extended to define the weak derivative of a distribution.

Reference: Weak derivative.

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