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This paper introduces a new concept of Boolean derivatives as a fusion of partial derivatives of Boolean functions (PDBFs). Three efficient algorithms for the calculation of PDBFs are presented. It is shown that Boolean function derivatives are useful for the application of identifying the location of edge pixels in binary images. The same concept is extended to the development of a new edge detection algorithm for grayscale images, which yields competitive results, compared with those of traditional methods. Furthermore, a new measure is introduced to automatically determine the parameter values used in the thresholding portion of the binarization procedure. Through computer simulations, demonstrations of Boolean derivatives and the effectiveness of the presented edge detection algorithm, compared with traditional edge detection algorithms, are shown using several synthetic and natural test images. In order to make quantitative comparisons, two quantitative measures are used: one based on the recovery of the original image from the output edge map and the Pratt's figure of merit.