Editing Weak derivative (section)

{{Short description|Generalisation of the derivative of a function}}
{{more footnotes|date=May 2014}}

In [[mathematics]], a '''weak derivative''' is a generalization of the concept of the [[derivative]] of a [[function (mathematics)|function]] (''strong derivative'') for functions not assumed [[Differentiable function|differentiable]], but only [[Integrable function|integrable]], i.e., to lie in the [[Lp space|L<sup>''p''</sup> space]] <math>L^1([a,b])</math>.

The method of [[integration by parts]] holds that for smooth functions <math>u</math> and <math>\varphi</math> we have

:<math display=block>\begin{align}
   \int_a^b u(x) \varphi'(x) \, dx 
   & = \Big[u(x) \varphi(x)\Big]_a^b - \int_a^b u'(x) \varphi(x) \, dx. \\[6pt]
 \end{align}</math>

A function ''u''<nowiki/>' being the weak derivative of ''u'' is essentially defined by the requirement that this equation must hold for all smooth functions <math>\varphi</math> vanishing at the boundary points (<math>\varphi(a)=\varphi(b)=0</math>).