Suppose that we have a sequence of functions $u_j$ that are in $L^{\infty}(0,1)$. Then the sequence of maps $N_j(s) := \u_j(s)\^2$ are also in $L^{\infty}(0,1)$. Hence they give rise to distributions and therefore has a distributional derivative. What is the explicit formula for $DN_j$? Is it related to the classical formula $2\langle u_j , Du_j\rangle$?

First, I do not understand why do you need a sequence of functions when the question involves an individual function. Suppose that $u$ is real valued. Then the product of the distributions $u$ and $u'$ may not even be defined. (This is the case when $u$ is the Heaviside function.) However, if the distributional derivative of $u$ is Lebesgue integrable, then $$ \frac{d}{dt}(\; u^2\;) = 2u u'. $$ 

