We analyze the relationship between the noise level of a function and the accuracy and reliability of derivatives and difference estimates. We derive and empirically validate measures of quality for both derivatives and difference estimates. Using these measures, we quantify the accuracy of derivatives and differences in terms of the noise level of the function. An interesting consequence of these results is that the derivative of a function is not likely to have working precision accuracy for functions with modest levels of noise.
Argonne Mathematics and Computer Science Division Preprint ANL/MCS-P2067-0312, April 2012.