= Differentiability of distribution translations
{title2=$\frac{d}{dh}\tau_hu=-\tau_hu'$}
With <translation of a distribution> defined by $\langle\tau_hu,\varphi\rangle=\langle u,\varphi(\mathord\cdot+h)\rangle$, differentiation of the <test function> gives $\frac d{dh}\tau_hu=-\tau_hu'$ in <weak convergence of distributions>. The <Taylor theorem> shows convergence in the <space of test functions>, including a common <compact support>. Thus $(\tau_{-h}u-u)/h\to u'$; fixing the translation convention prevents a sign error.
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