Accurate (\ensuremath{\lesssim}1%) predictions for the anisotropy of the cosmic background radiation (CBR) are essential for using future high-resolution (\ensuremath{\lesssim}1\ifmmode^\circ\else\textdegree\fi{}) CBR maps to test cosmological models. In many inflationary models the variation (``running'') of the spectral index of the spectrum of density perturbations is a significant effect and leads to changes of around 1\char21{}10 % in the CBR power spectrum. We propose a general method for taking running into account which uses the derivative of the spectral index (dn/dlnk). Conversely, high-resolution CBR maps may be able to determine dn/dlnk, giving important information about the inflationary potential.