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Accurate ($\lesssim 1\% $) predictions for the anisotropy of the CosmicBackground Radiation (CBR) are essential for using future high-resolution($\lesssim 1^\circ$) CBR maps to test cosmological models. In many inflationarymodels the variation (``running'') of the spectral index of the spectrum ofdensity perturbations is a significant effect and leads to changes of around1\% to 10\% in the CBR power spectrum. We propose a general method for takingrunning into account which uses the derivative of the spectral index ($dn/d\lnk$). Conversely, high-resolution CBR maps may be able to determine $dn/d\ln k$,giving important information about the inflationary potential.Comment: Discussion of calculation clarified; error corrected which reduces  estimated effect for chaotic inflatio

CBR anisotropy and the running of the scalar spectral index