Weak derivative

In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable, but only integrable, i.e. to lie in the L^p space ${\displaystyle \mathrm {L} ^{1}([a,b])}$. See distributions for an even more general definition.

Let ${\displaystyle u}$ be a function in the Lebesgue space ${\displaystyle L^{1}([a,b])}$. We say that ${\displaystyle v}$ in ${\displaystyle L^{1}([a,b])}$ is a weak derivative of ${\displaystyle u}$ if,

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